Aug 31, 2017 - Klaus Adam, University of Mannheim & CEPR ...... Princeton University Press, Princeton. ...... Transi
Discussion Paper Deutsche Bundesbank No 25/2017 Optimal trend inflation Klaus Adam (University of Mannheim and CEPR)
Henning Weber (Deutsche Bundesbank)
Discussion Papers represent the authors‘ personal opinions and do not necessarily reflect the views of the Deutsche Bundesbank or its staff.
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Non-technical Summary Research Question How high is the optimal inflation rate, i.e., the rate of price increase with the least distortionary effect on relative prices? While this is a stock question in macroeconomics, there is one factor thus far that has been left out of attempts to analyse the optimal inflation rate: firm-level productivity trends. In this paper, we focus on these trends and argue that productivity trends experienced by firms and their products across the life cycle impact on the optimal inflation rate.
Contribution We present a sticky-price model incorporating heterogeneous firms and systematic firmlevel productivity trends and show that this model delivers very different predictions for the optimal inflation rate than canonical sticky price models featuring homogenous firms. Furthermore, we use detailed micro data from the US Census Bureau to estimate the inflation-relevant productivity trends at the firm level and quantify the optimal US inflation rate.
Results Our model predicts that the optimal steady-state inflation rate generically differs from zero and inflation optimally responds to productivity disturbances that affect firm-level productivity trends. Our estimation based on the micro data implies that the optimal US inflation rate is positive. It was slightly above 2 percent in the year 1986, but continuously declined thereafter, reaching about 1 percent in the year 2013. Our analysis thus indicates that structural change and structural policy, which impact on firm-level technological progress, can have a prolonged impact on the optimal inflation rate.
Nichttechnische Zusammenfassung Fragestellung Wie hoch ist die optimale Inflationsrate? Die Frage, welche Preissteigerungsrate die relativen Preise am wenigsten verzerrt, ist ein Klassiker der Makro¨okonomie. Allerdings blieb bisher ein Faktor bei der Analyse der optimalen Inflationsrate unbeachtet: Produktivit¨atstrends auf Unternehmensebene. In unserer Studie konzentrieren wir uns auf diese Trends und argumentieren, dass Produktivit¨atstrends, die Unternehmen und ihre G¨ uter u ¨ber den Lebenszyklus durchlaufen, einen Einfluss auf die optimale Inflationsrate haben.
Beitrag Wir betrachten einen Modelrahmen mit nominaler Rigidit¨at und systematischen Produktivit¨atstrends auf Unternehmensebene. Die Implikationen dieses Modelrahmens unterscheiden sich stark von den Implikationen weitverbreiteter makro¨okonomischer Modelle mit nominaler Rigidit¨at, aber homogenen Unternehmen. Anhand eines detaillierten Datensatzes f¨ ur die USA, der Informationen zur Besch¨aftigung auf Unternehmensebene f¨ ur alle privaten Unternehmen bereitstellt, sch¨atzen wir die inflations-relevanten Produktivit¨atstrends auf Unternehmensebene und quantifizieren die optimale Inflationsrate.
Ergebnisse Unser Modellrahmen impliziert, dass sich die optimale, langfristig Inflationsrate generell von Null unterscheidet und dass Ver¨anderungen derProduktivit¨atstrends auf Unternehmensebene auch die optimale Inflationsrate im Laufe der Zeit ver¨andern. Unsere auf den Unternehmensdaten basierende Sch¨atzung f¨ ur die USA ergibt, dass die optimale Inflationsrate klar positiv ist, aber seit Mitte der 80er Jahre von etwas u ¨ber zwei Prozent auf mittlerweile circa ein Prozent gefallen ist. Unsere Analyse legt daher nahe, dass Strukturver¨anderungen und Strukturpolitik, die den technologischen Fortschritt auf Unternehmensebene beeinflussen, einen lang anhaltenden Einfluss auf die optimale Inflationsrate haben k¨onnen.
Bundesbank Discussion Paper No 25/2017
Optimal Trend In‡ation Klaus Adam, University of Mannheim & CEPR Henning Weber, Deutsche Bundesbank July 12, 2017
Abstract We present a sticky-price model incorporating heterogeneous …rms and systematic …rm-level productivity trends. Aggregating the model in closed form, we show that it delivers radically di¤erent predictions for the optimal in‡ation rate than canonical sticky price models featuring homogenous …rms: (1) the optimal steadystate in‡ation rate generically di¤ers from zero and (2) in‡ation optimally responds to productivity disturbances. Using micro data from the US Census Bureau to estimate the in‡ation-relevant productivity trends at the …rm level, we …nd that the optimal US in‡ation rate is positive. It was slightly above 2 percent in the year 1986, but continuously declined thereafter, reaching about 1 percent in the year 2013. Keywords: optimal in‡ation rate, sticky prices, …rm heterogeneity JEL Class. No.: E52, E31, E32
We would like to thank Matthias Kehrig for generously supporting us through access to the LBD database. We also thank Pierpaolo Benigno, Saki Bigio, V.V. Chari, Argia Sbordone, Stephanie SchmittGrohé, Michael Woodford, seminar participants at the 2016 CSEF-CIL-UCL conference in Alghero, the 12th Dynare conference in Rome, 2016, the 2016 EUI Workshop on Economic Policy Challenges, Florence, the 2016 German Economic Association (VfS) meeting, and the 2017 EABCN meeting at the BdF for comments and suggestions. The opinions expressed in this paper are those of the authors and do not necessarily re‡ect the views of the Deutsche Bundesbank, the Eurosystem, or their sta¤. Any remaining errors are our own.
1
Introduction
This paper introduces heterogeneous …rms and empirically plausible …rm-level productivity trends into an otherwise standard sticky-price economy. It shows that some of the most fundamental implications of canonical sticky-price models with homogeneous …rms fail to survive within such a generalized sticky-price setup. The optimal steady-state in‡ation rate generically di¤ers from zero and in‡ation optimally responds to productivity disturbances, unlike in settings with homogeneous …rms. Moreover, the paper documents that the predictions of the homogeneous …rm model turn out to be non-robust in the sense that they are discontinously a¤ected by the presence of …rm heterogeneity. We thus present an example in which microeconomic heterogeneity matters for macroeconomic policy prescriptions, an issue that has received renewed interest recently (Ahn et al. (2017), Kaplan and Violante (2014)). Due to the technical di¢ culties associated with aggregating heterogeneous …rm models, it is standard in the sticky-price literature to abstract from all …rm-level heterogeneity beyond that generated by price adjustment frictions themselves. As is well known, price adjustment frictions then tightly anchor the optimal steady-state in‡ation rate at zero, e.g., Woodford (2003).1 As we show, this rather robust but somewhat puzzling implication of standard sticky-price models arises precisely because of the homogeneity assumption. Homogeneity implies that the productivity of price-adjusting …rms equals that of non-adjusting …rms. With economic e¢ ciency requiring relative prices to re‡ect relative productivities, it calls for price-adjusting …rms to charge the same price as charged on average by non-adjusting …rms, i.e., it calls for zero in‡ation.2 The present paper extends the basic sticky-price setup by introducing …rm heterogeneity and systematic …rm-level productivity trends. Such …rm-level trends are clearly present in micro data, but are routinely abstracted from in the sticky price literature. New …rms, for example, tend to be initially small, i.e., tend to be initially unproductive when compared to existing …rms.3 Some of the young …rms become more productive over time and grow, others become unproductive and exit the economy. We show how such life-cycle related productivity dynamics cause the average productivity of price-adjusting …rms to generally di¤er from the average productive of non-adjusting …rms. Economic e¢ ciency then requires that adjusting …rms set on average di¤erent prices than existing …rms, which causes in‡ation or de‡ation to be optimal in steady state. We show this by aggregating the non-linear sticky price model with heterogeneous …rms in closed form and 1
Section 2 discusses a range of extensions of the basic framework considered in the literature and their implications for the optimal in‡ation rate. 2 Yun (2005) shows, using a setting with homogeneous …rms, that if initial prices do not re‡ect initial productivities, the optimal in‡ation rate can display deterministic transitory deviations from zero. 3 This does not rule out that new …rms are in age-adjusted terms more productive than old …rms. Our setup will allow for this possibility.
1
by deriving analytical expressions for the optimal in‡ation rate. The heterogeneous …rm model that we present is formulated in abstract terms and allows for a variety of economic interpretations through which …rm heterogeneity arises. One interpretation is - as alluded to above - that heterogeneity arises from …rm entry and exit and the associated life-cycle dynamics of …rm productivity. This is also the interpretation that we shall consider in our empirical analysis. Yet, as explained in the main text, the model can equally be interpreted as one in which heterogeneity arises from product substitution or product quality improvements. To …x ideas, consider a sticky-price model with Calvo type or menu-cost type price adjustment frictions in which a measure
0 of randomly chosen …rms becomes un-
productive and exits the economy. Exiting …rms are replaced by a measure
of young
new …rms. Our setup then features three systematic productivity trends, each of which has di¤erent implications for the optimal in‡ation rate. First, there is a common trend in total factor productivity (TFP), which a¤ects all …rms equally. The common TFP trend captures general-purpose innovations that are adopted by all …rms simultaneously. As in a standard homogeneous-…rm model, it does not a¤ect the optimal in‡ation rate. Second, there is an experience trend in …rm TFP, which determines how …rms accumulate experience with age. The experience trend may capture productivity gains from learningby-doing or other forms of experience accumulation. As we show, this productivity trend generates a force towards positive in‡ation rates. Third, there is a cohort productivity trend, which determines the productivity level of newly entering …rms. This trend captures the fact that new …rms tend to bring new technologies into the economy that are not (yet) used by other …rms.4 The cohort trend will be a force towards making de‡ation optimal. Taken together, the optimal steady-state in‡ation rate in our setting depends on the strength of the experience trend relative to the strength of the cohort trend, whenever there is some positive …rm turnover ( > 0). The optimal steady-state in‡ation rate is itself independent of the …rm turnover rate, as long as
> 0. Yet, in the absence of
…rm turnover ( = 0), the optimal steady-state in‡ation rate collapses to zero, i.e., to the optimal in‡ation rate of a homogeneous …rm model. It is in this sense, that the in‡ation predictions of the homogeneous …rm model turn out to be non-robust. To provide economic intuition for these …ndings, consider two polar settings. The …rst setting abstracts from the presence of a cohort trend and considers a setting where the only trend is that …rms accumulate experience over time.5 If an old …rm becomes unproductive and exits the economy, the new …rm that replaces it will not have accumulated any 4
Newly entering …rms are endowed with the cohort productivity level, in addition to the common TFP component, and then gradually accumulate experience over time. 5 As mentioned before, we can abstract from the common TFP trend, as it does not a¤ect the optimal in‡ation rate.
2
experience yet. The new …rm will thus be less productive than the remaining set of old …rms.6 From a welfare standpoint, the optimal price of new …rms should therefore exceed the average price of existing …rms, so as to accurately re‡ect relative productivities. Achieving this requires either that new …rms choose higher prices or that old …rms reduce prices, or a combination thereof. In the presence of sticky prices, price reductions by old …rms are costly. In timedependent adjustment models, they lead to ine¢ cient price dispersion due to asynchronous price adjustment; in state-dependent pricing models, they require …rms to pay adjustment costs. Therefore, it is optimal to implement the e¢ cient relative price exclusively by having new …rms charge higher prices, while all other …rms hold their prices steady. Clearly, this implies that the aggregate in‡ation rate must be positive in the steady state. Now consider the second polar setting, in which there is no experience e¤ect and the only trend is a positive cohort trend. New …rms are then more productive than the existing set of old …rms, thus optimally charge lower prices than existing …rms. This makes negative rates of in‡ation optimal.7 We also determine in closed form the dynamic response of the optimal in‡ation rate following shocks to experience and cohort productivity. We show that such shocks have fairly persistent e¤ects on the optimal in‡ation rate, especially in settings in which positive but close to zero. A low value for
is
causes a persistent shock to the experience
level of existing …rms to give rise to a persistent change in relative productivity between existing …rms and new entrants. Likewise, a persistent shock to the productivity level of new cohorts causes persistent relative productivity di¤erences between existing and new …rms. These persistent productivity di¤erences require that in‡ation also moves persistently along the transition until the productivity distribution has again reached its steady state. Importantly, one cannot infer the in‡ation-relevant cohort and experience trends by observing aggregate productivity. As we show, these trends are not identi…ed because in‡ation-neutral TFP trends mask the underlying in‡ation-relevant cohort and experience trends at the aggregate level. In our empirical analysis we thus resort to micro evidence on …rm dynamics. To obtain a plausible framework for empirical analysis, we extend our baseline setting to a multi-sector economy. The multi-sector setup allows for sector-speci…c experience and cohort trends, sector-speci…c "common" TFP trends, as well as sector-speci…c …rm turnover rates and degrees of price stickiness. We then show that the in‡ation rate that maximizes steady-state welfare is a weighted average of the in‡ation rates that would 6
The new …rm will be in age-adjusted terms more productive than all old …rms, once one allows for a positive cohort trend. 7 Due to price-setting frictions, it is again not optimal that old …rms adjust prices.
3
achieve e¢ cient relative prices within each sector individually.8 Based on this insight, we devise a model-based empirical strategy that allows us to estimate these sector-speci…c cohort and experience trends and thus the optimal in‡ation rate from …rm-level data. To estimate the relevant …rm-level trends, we use the Longitudinal Business Database (LBD) of the US Census Bureau, which covers all private sector establishments in the United States, and estimate the relevant cohort and experience trends and their evolution over time. Our regression results show that the optimal in‡ation rate implied by our model is positive but approximately halved over the period 1986 to 2013. Depending on the precise value of the elasticity of product demand assumed, the level of the optimal in‡ation rate varies. For our preferred demand elasticity speci…cation, the optimal in‡ation rate declined from around 2% in 1986 to approximately 1% in 2013. The remainder of the paper is structured as follows. Section 2 discusses the related literature. Section 3 presents our heterogeneous …rms model with sticky prices. Section 4 analytically aggregates the model, and section 5 shows that the ‡exible-price equilibrium is …rst best when a Pigouvian output subsidy corrects …rms’monopoly power. The main results on the optimal rate of in‡ation for the non-linear model are presented in closedform in section 6. Section 7 discusses the implications of the main results for the optimal steady-state in‡ation rate and steady-state welfare. It also shows how the optimal in‡ation rate jumps discontinuously when moving from a standard sticky-price economy ( = 0) to one including …rm turnover ( > 0). Section 8 determines the utility costs of implementing suboptimal in‡ation and section 9 discusses the optimal response of the in‡ation rate to economic disturbances. Section 10 extends the baseline setup to a multi-sector economy, allowing for a considerable degree of sectoral heterogeneity. Using a model-consistent approach, section 11 estimates the optimal in‡ation rate for the US economy using the LBD data. Section 12 discusses the robustness of our …ndings towards various extensions. A conclusion brie‡y summarizes. Proofs and technical material are relegated to a series of appendices.
2
Related Literature
Only few papers discuss the relationship between the optimal in‡ation rate and productivity trends. All of these focus on aggregate or sectoral productivity trends and …nd that the optimal in‡ation rate is (slightly) negative in their calibrated models. Amano et al. (2009), for instance, consider an economy with aggregate productivity growth and sticky wages and prices. They show how monetary policy a¤ects wage and price mark-ups and that this can make it optimal to implement de‡ation, so as to reduce wage mark-ups. 8
The in‡ation rate that achieves e¢ ciency in a speci…c sector again depends on the sector-speci…c cohort and experience trends.
4
Wolman (2011) considers a two-sector sticky-price economy with sectoral productivity trends. He shows that - even in the absence of monetary frictions - the optimal in‡ation rate is either negative or close to zero, depending on the precise modeling of price adjustment frictions. Golosov and Lucas (2007) and Nakamura and Steinsson (2010) consider sticky-price setups with heterogeneous …rms and study monetary non-neutrality within these setups. They do not consider the issue of the optimal in‡ation rate. Firms in their settings are subject to random idiosyncratic productivity shocks. This di¤ers from the present setup which features idiosyncratic shocks that give rise to systematic productivity adjustments (as implied by the cohort and experience trends). The idiosyncratic nature of productivity shocks in Golosov and Lucas (2007) and Nakamura and Steinsson (2010) causes …rms with very positive or very negative idiosyncratic productivity shocks to adjust prices. The productivity of price-adjusting …rms is thus on average similar to that of non-adjusting …rms, suggesting zero in‡ation to be optimal. The present paper is also related to a large literature studying the determinants of optimal in‡ation, most of which …nds that the optimal in‡ation rate is either negative or close to zero. None of these papers makes a connection between the optimal in‡ation rate and …rm-level productivity dynamics. In classic work, Kahn, King and Wolman (2003) explore the trade-o¤ between price adjustment frictions, which call for price stability, and monetary frictions, which call for a Friedman-type de‡ation. They demonstrate how a slight rate of de‡ation is optimal in such frameworks. In a comprehensive survey, Schmitt-Grohé and Uribe (2010) document the robustness of these …ndings to a large number of natural extensions. They show that taxation motives, including the presence of untaxed income, foreign demand for domestic currency (Schmitt-Grohé and Uribe (2012a)), as well as a potential quality bias in measured in‡ation rates (Schmitt-Grohé and Uribe (2012b)), are all unable to rationalize signi…cantly positive rates of in‡ation. Adam and Billi (2006, 2007) and Coibion, Gorodnichenko and Wieland (2012) explicitly incorporate a lower bound on nominal interest rates into sticky-price economies. They …nd that fully optimal monetary policy is consistent with close to zero average rates of in‡ation. While zero lower bound episodes make it optimal to promise in‡ation in the future, these promises should only be made conditionally on being at the lower bound, which happens rather infrequently; see Eggertsson and Woodford (2003) for an early exposition. A number of papers …nd positive in‡ation rates to be optimal on average when introducing downward nominal wage rigidities into the standard setup. Kim and Ruge-Murcia (2009) argue that such rigidities allow optimal in‡ation rates of approximately 0.35% on average to be justi…ed when using a model with aggregate shocks only. Looking at a
5
setting with idiosyncratic shocks, Benigno and Ricci (2011) also …nd a positive steadystate in‡ation rate to be optimal.9 Carlsson and Westermark (2016) consider a setting with nominal wage rigidities and search and matching frictions in the labor market. They show how a standard US calibration of the model implies failure of the Hosios condition and justi…es an annual in‡ation rate of about 1.16%. Schmitt-Grohé and Uribe (2013) analyze the case for temporarily elevated in‡ation in the euro area due to the presence of downward rigidity of nominal wages. Brunnermeier and Sannikov (2016) show that the optimal in‡ation rate can also be positive in a model without nominal rigidities. They present a model with undiversi…able idiosyncratic capital income risk in which the optimal in‡ation rate increases with the amount of idiosyncratic risk. There is also a literature studying endogenous …rm entry decisions in homogeneous …rm economies, focusing on the e¤ect of in‡ation on the …rm entry margin, e.g., Bergin and Corsetti (2008), Bilbiie et al. (2008) and Bilbiie, Fujiwara and Ghironi (2014). Bilbiie et al. (2014) document - amongst other things - that the welfare optimal in‡ation rate is positive whenever the bene…t of additional varieties to consumers falls short of the market incentives for creating these varieties. In‡ation then reduces the value of creating varieties and brings …rm entry closer to its e¢ cient (lower) level. The present paper abstracts from endogenous …rm entry decisions and thus from the implication of monetary policy for the entry margin, instead considers a setting with heterogeneous …rms in which entry and exit is driven by exogenous productivity dynamics. A set of empirical papers decompose the observed US in‡ation rate into a trend and cyclical component and shows that trend in‡ation displays substantial low-frequency variation over time, e.g., Cogley and Sargent (2001), Cogley, Primiceri and Sargent (2010). The sticky-price literature has reacted to these facts by incorporating trend in‡ation into their workhorse models; see Ascari and Sbordone (2014) and Cogley and Sbordone (2008). Trend in‡ation emerges in these setups because the central bank pursues an exogenous in‡ation target, which is non-zero and potentially time-varying. Primiceri (2006), Sargent (1999), and Sargent, Williams and Zha (2006) present settings in which policymakers learn about the Phillips curve trade-o¤ and show how this can endogenously give rise to the observed low-frequency movements in US in‡ation. The present paper does not explore to what extent changes in …rm-level productivity can contribute to explaining the observed US in‡ation history, as it focuses on the normative implications of these trends. Exploring the positive content of the theory presented in this paper appears to be an interesting avenue for further research. 9
Since positive in‡ation has no welfare costs in their setup, they do not quantify the optimal in‡ation rate.
6
3
Economic Model
We consider a cashless economy with nominal rigidities and monopolistically competitive …rms. The model is entirely standard, except for the more detailed modeling of …rmlevel productivity and price adjustment dynamics. Speci…cally, we augment the standard sticky-price setup by idiosyncratic …rm-level productivity adjustments that arrive in conjunction with a price adjustment opportunity. This gives rise to a setting with heterogeneous …rm-level productivities in which the productivity of price-adjusting …rms is not necessarily equal to that of non-adjusting …rms. For simplicity, we derive our results within a time-dependent price adjustment model à la Calvo (1983). As we argue in section 12.1, our main …ndings remain unaltered if we look instead at a setting where price adjustment frictions take the form of menu costs. The next section introduces our generalized …rm setup in abstract terms. Section 3.2 provides alternative economic interpretations of the setup.
3.1
Technology, Prices and Price Adjustment Opportunities
Each period t = 0; 1; : : : there is a unit mass of monopolistically competitive …rms indexed by j 2 [0; 1]. Each …rm j produces output Yjt , which enters as an input into the production
of an aggregate consumption/investment good Yt according to R1
Yt = where 1
0, respectively. Productivity dynamics in the previous setting feature three trends: (1) the common growth trend at ; (2) the experience growth trend gt , which applies in the absence of shocks; and (3) the productivity growth trend qt , which determines the e¤ects of -shocks on technology. Each of these three growth trends has a di¤erent implication for the optimal in‡ation rate. To understand the productivity dynamics implied by the previous setup, consider …rst the special case with = 0. In the absence of idiosyncratic -shocks to …rm technology, all …rms experience the same productivity growth rate at gt . Such a setting with homogeneous productivity growth across all …rms is the one routinely considered in the sticky-price literature.11 Next, consider the case
> 0 and let sjt denote the number of periods that have elapsed since …rm j last experienced a -shock (i.e., j;t sjt = 1 and jet = 0 for e t = sjt + 1; :::; t). Firm-speci…c productivity Zjt in equation (4) can then be written as
t
Zjt = Gjt Qt 10
sjt ;
In the absent of aggregate technology growth, the formulation of …xed costs in equation (3) corresponds to that used in Melitz (2003). 11 For the case = 0, our setting still allows for a non-degenerate initial distribution of …rm productivities. Typically, this initial distribution is also assumed to be degenerate in the sticky-price literature. As we show below, the additional assumption of a degenerate initial distribution is not key for the conclusion that zero in‡ation is optimal, as long as initial prices re‡ect initial productivities, see Yun (2005) for a discussion of this and related issues in a homogeneous …rm setting.
8
where Gjt =
(
for sjt = 0
1 gt Gjt
1
otherwise,
and where Qt follows equation (5). This alternative formulation illustrates that all …rms hit by a -shock in t upgrade idiosyncratic productivity to Zjt = Qt , so that Qt can be interpreted as capturing a "cohort e¤ect" of productivity dynamics, where cohorts are determined by the arrival time of the last -shock. Following any -shock, the …rm experiences productivity gains, as described by the process Gjt , as long as no further shocks arrive. Since the productivity gains Gjt are lost with the arrival of the next -shock, one can interpret the process Gjt as capturing "experience" or "learning-by-doing e¤ects" associated with the cohort production technology Qt
sjt .
Following a -shock in period t,
our speci…cation thereby implies that …rm productivity increases (temporarily decreases) if Qt has been growing faster (slower) than Gjt since the time of arrival of the last -shock prior to period t. Note, however, that as long as Qt displays a positive growth trend (q > 0), …rms always become more productive over time in experience-adjusted terms, even if Qt grows slower than Gjt . Indeed, in our setting the long-term growth rate of …rms’ productivity is determined by the process at qt , as the experience growth rate gt generates - due to the occasional reset - only temporary level e¤ects for …rm productivity. As usual, we de…ne the aggregate price level as Z 1 Pjt1 dj Pt
1 1
:
(6)
0
Cost minimization in the production of …nal output Yt implies Z 1 Yjt Pt = Pjt dj, Yt 0
which shows that the price level is an expenditure-weighted average of the prices in the di¤erent expenditure categories, in line with the practice at statistical agencies. Following the approach of the Bureau of Labor Statistics, we furthermore assume that all current product versions enter the computation of the CPI and thus the in‡ation rate.12 The in‡ation rate is de…ned as t
Pt =Pt 1 :
12
The section "Item replacement and quality adjustment" in chapter 17 of the BLS Handbook of Methods, BLS (2015), describes how the changeover of discontinued product versions is handled. If a data collector cannot …nd anymore a product version that was previously contained in the sample, she/he replaces it with a new version. The price of the old version enters the previous price index and the price of the new version enters the current price index. The BLS also seeks to adjust for quality di¤erences across versions. Armknecht et al. (1996) shows that about 3% of products are discontinued each month and table 9.2 shows that more than 50% of the replacement versions fall into the category "direct comparisons", for which no quality adjustment is made; for the remaining replacements there is either a direct quality adjustment or quality adjustment is imputed via di¤erent methods. As will become clear in section 3.2, our setup will be consistent with statistical agencies making such quality adjustments.
9
Figure 1: Productivity dynamics in a setting with …rm entry and exit
We also assume that at = a at , qt = q qt , and gt = g
g t
with
a q g t; t; t
> 0 being stationary
shocks with an arbitrary contemporaneous and intertemporal covariance structure, satisfying E[ at ] = E[ qt ] = E[ gt ] = 1. To obtain a well-de…ned steady state and to insure that relative prices in the ‡exible-price economy remain bounded, we assume throughout the paper (1
3.2
) (g=q)
1
< 1:
(7)
Alternative Interpretations of the Firm Setup
The previous section de…ned -shocks (
jt
= 1) as an idiosyncratic change in …rm-level
productivity that is associated with a price adjustment opportunity. This section presents three alternative economic interpretations of -shocks that highlight alternative economic sources of …rm heterogeneity and that explain why productivity changes may plausibly be associated with price ‡exibility at the …rm level. Firm entry and exit. It is possible to interpret -shocks as a …rm exit and entry event. Indeed, this is the interpretation that we will adopt in our empirical application of the model in section 11. Speci…cally, the event
jt
= 1 can be interpreted as an event in
which …rm j becomes permanently unproductive and thus exits the economy. Each exiting 10
…rms is then replaced by a newly entering …rm to which we assign for simplicity the same …rm index j. The variable Qt then captures the productivity level of the cohort of …rms that enters in period t, and Gjt captures the experience accumulated over the lifetime of the …rm. The assumption that …rm prices are ‡exible following a -shock should then be interpreted as newly entering …rms being able to freely choose the price of their product. It is worth noting that …rm entry and exit rates are high in the United States and range in the order of 8-12% per year, see …gure 3 in Decker et al. (2014). Figure 1 illustrates the …rm level productivity dynamics for the empirically plausible setting in which the cohort trend is positive (q > 0), but less strong than the experience trend (g > q). To simplify the exposition, the …gure depicts the deterministic dynamics and abstracts from the TFP trend a, which does not a¤ect the distribution of relative productivities across …rms. The line Qt in the …gure indicates the cohort trend and captures the productivity of newly entering …rms at each point in time. The lines starting at the cohort trend line capture the productivity dynamics of the entering cohorts over time. Since g > q, the productivity of existing …rms grows faster than that of new entrants, so that existing …rms are initially more productive and thus larger than newly entering …rms. In experience-adjusted terms, however, newly entering …rms are the most productive …rms in the economy. The downward-pointing dashed arrows indicate the productivity losses of exiting …rms that have been hit by a -shock. For simplicity, the …gure assumes that their productivity permanently drops to zero. As should be clear from the …gure, the entry and exit dynamics imply an exponential distribution for …rm age. Coad (2010) shows that such an age distribution is empirically plausible and how it generates, together with (productivity) growth shocks, a Pareto distribution for …rm size, in line with the observed …rm size distribution. Product substitution. The event
jt
= 1 can also be interpreted as an event in
which the product previously produced by …rm j is no longer demanded by consumers. Firm j reacts to this by introducing a new product, which - for simplicity - is assigned the same product index j. The variable Qt then captures the productivity level associated with products that are newly introduced in t and Gjt captures experience accumulation in producing the new product. Product substitutions, e.g., in the form of new product versions or models, take place rather frequently in the data and are also prevalent in the CPI baskets of statistical agencies (see section III.C in Nakamura and Steinsson (2008) for evidence on the rate of product substitution in the US CPI). Evidence provided in Moulton and Moses (1997), Bils (2009) and Melser and Syed (2016) furthermore shows that the prices of new products are typically higher than those that they replace, even after accounting for quality improvements.13 It thus appears reasonable to assume price 13
Evidence provided in Bils (2009) shows that in‡ation for durables ex computers over the period 1988-2006 averaged 2.5% per year, but when including only matched items, the in‡ation rate was -3.7%
11
‡exibility for new products. Quality improvements. Let Qjt denote the quality of the product produced by …rm j in period t. De…ning Qjt = Qt
sjt ,
the event
jt
= 1 captures the situation in which
…rm j upgrades the quality of its product from level Qt
1 sj;t
1
to level Qt . Let aggregate
output produced with intermediate inputs of di¤erent quality be given by Yt =
Z
1
1
Qjt Yejt
0
1
dj
;
and let …rm j’s output of quality level Qjt be given by 1 Yejt = At Gjt Kjt
1
1
Ljt
Ft ;
where Gjt now captures experience e¤ects associated with producing quality Qjt . Finally, let Pejt denote the price of a unit of good j of quality level Qjt . Assuming that statistical agencies perfectly adjust the price level for quality changes over time, we have 111 0 Z 1 e !1 Pjt : djA Pt = @ Qjt 0
As is easily veri…ed, this setup with quality improvements is mathematically identical to the one with productivity changes spelled out in the previous section.14 Again, as in the case with product substitution, it appears natural to assume that …rms can ‡exibly price goods featuring improved quality features.
3.3
Optimal Price Setting
Firms choose prices, capital and hours worked to maximize pro…ts. While price adjustment is subject to adjustment frictions, factor inputs can be chosen ‡exibly. Letting Wt denote the nominal wage and rt the real rental rate of capital, …rm j chooses the factor input mix so as to minimize production costs Kjt Pt rt + Ljt Wt subject to the constraints imposed by the production function (2). Let Ijt
Ft + Yjt =(At Qt 1
1
1
denote the units of factor inputs (Kjt
sjt Gjt )
Ljt ) required to produce Yjt units of output. As
appendix A.1 shows, cost minimization implies that the marginal costs of Ijt are given by M Ct =
Wt 1=
1
Pt rt 1 1=
1
1
:
(8)
per year. 14 The quality-adjusted price Pejt =Qjt and the quality-adjusted quantity Yejt Qjt then correspond to the price Pjt and quantity Yjt , respectively, in the previous section.
12
Now consider a …rm in period t that can freely choose its price because it has experienced either a -shock or a Calvo adjustment shock. Letting
denote the probability implied
by the Calvo process that the …rm has to keep its price (0 < able to reoptimize its price with probability (1
< 1), the …rm will not be
) at any future date, i.e., whenever it
undergoes neither a - shock nor a Calvo adjustment shock.15 The price-setting problem of a …rm that can optimize its price in period t is thus given by max Et Pjt
1 X
))i
( (1
i=0
t;t+i
Pt+i
[(1 + )Pjt+i Yjt+i
s:t: Ijt+i = Ft+i + Yjt+i =At+i Qt Yjt+i = (Pjt+i =Pt+i ) Pjt+i+1 = where
(9)
M Ct+i Ijt+i ]
sjt Gjt+i ;
Yt+i ;
t+i;t+i+1 Pjt+i :
denotes a sales tax/subsidy and
t;t+i
denotes the representative household’s
discount factor between periods t and t + i. The …rst constraint captures the …rm’s technology, the second constraint captures the demand function faced by the …rm, as implied by equation (1), and the last constraint captures how the …rm’s price is indexed over time (if at all) in periods in which prices are not reset optimally. We consider general price indexation schemes and allow
to be a function of aggregate variables up
t+i;t+i+1
to period t + i.16 In the absence of indexation, we have Appendix A.2 shows that the optimal price Pjt? Pt
Qt
sjt Gjt
Qt
Pjt?
= 1 for all i
0.
can be expressed as 1 11+
=
t+i;t+i+1
Nt ; Dt
(10)
where the variables Nt and Dt are functions of aggregate variables only and evolve recursively according to M Ct Nt = + (1 Pt At Qt Dt = 1 + (1
)Et
)Et "
"
Yt+1 ( t;t+1 Yt
Yt+1 ( t;t+1 Yt
1
t;t+1 )
t;t+1 )
Pt+1 Pt
Pt+1 Pt 1
qt+1 gt+1 #
Dt+1 :
Nt+1
#
(11) (12)
Equation (10) shows that the optimal reset price of a …rm depends only on how its own productivity (At Qt
sjt Gjt )
relates to the productivity of a …rm hit by a -shock in period
15
In any period, the …rm can adjust its price with probability due to the occurrence of a -shock and with probability (1 )(1 ) due to the occurrence of a Calvo price adjustment shock. 16 We only require that price indexation is such that the price-setting problem remains well de…ned, that price indexation does not give rise to multiplicities of the optimal in‡ation rate and that indexation is such that t;t+1 = 1 in a steady state without in‡ation. For instance, when indexing occurs with respect to lagged in‡ation according to t;t+1 = ( t ) with 0, we rule out > 1 to avoid non-existence of optimal plans and rule out = 1 to avoid multiplicities of the steady-state in‡ation rate.
13
t (At Qt ), as well as on aggregate variables. It is precisely this feature which permits aggregation of the model in closed form. Equation (10) furthermore shows that more productive …rms optimally choose lower prices. For the special case with homogeneous …rms, where relative productivity is always equal to one (Qt
sjt Gjt =Qt
= 1), equations
(10)-(12) reduce to those capturing price dynamics in a standard homogeneous-…rm model.
3.4
Household Problem
There is a representative household with balanced growth consistent preferences given by ! 1 1 X [C V (L )] 1 t t t E0 ; t 1 t=0 where Ct denotes private consumption of the aggregate good, Lt labor supply, ence shock with E[ t ] = 1 and
2 (0; 1) the discount factor. We assume
t
a prefer-
> 0 and that
V ( ) is such that period utility is strictly concave in (Ct ; Lt ) and that Inada conditions are satis…ed. The household faces the ‡ow budget constraint Z 1 Wt Bt 1 Bt jt = (rt + 1 d)Kt + Lt + dj + (1 + it 1 ) Tt ; Ct + Kt+1 + Pt Pt Pt 0 Pt where Kt+1 denotes the capital stock, Bt nominal government bond holdings, it
1
the
nominal interest rate, Wt the nominal wage rate, rt the real rental rate of capital, d the depreciation rate of capital,
jt
nominal pro…ts from ownership of …rm j, and Tt lump
sum taxes. Household borrowing is subject to a no-Ponzi scheme constraint. The …rstorder conditions characterizing optimal household behavior are entirely standard and are derived in Appendix A.3. To insure existence of a well-de…ned balanced growth path, we assume throughout the paper that < (aq) :
3.5
Government
To close the model, we consider a government which faces the budget constraint Z 1 Bt 1 Pjt Bt = (1 + it 1 ) + Yjt dj Tt ; Pt Pt Pt 0
where
denotes a sales subsidy, which will be used to correct for the monopolistic distor-
tions in product markets. The government levies lump sum taxes Tt , so as to implement a bounded state-contingent path for government debt Bt =Pt .17 Since we consider a cashless limit economy, there are no seigniorage revenues, even though the central bank controls the nominal interest rate. We furthermore assume that monetary policy is not constrained by a lower bound on nominal interest rates. The equilibrium concept is standard and de…ned in appendix A.5. 17
The household’s transversality condition will then automatically be satis…ed in equilibrium.
14
4
Analytical Aggregation with Heterogeneous Firms
This section outlines the main steps that allow us to aggregate the model in closed form. In a …rst step, we derive a recursive representation describing the evolution of the aggregate price level Pt over time. In a second step, we derive a closed-form expression for the aggregate production function. In a last step, we show how to appropriately detrend aggregate variables, so as to render them stationary. Evolution of the aggregate price level. Let Pt? …rm that last experienced a -shock in t (s Qk
t k+j 1;t k+j
denote the optimal price of a
s and that has last reset its price in t ? t k;t Pt s;t k ,
0). In period t, this …rm’s price is equal to
k
j=1
s;t k
captures the cumulative e¤ect of price indexation (with
the absence of price indexation). Let raised to the power of 1
t k;t
t k;t
=
1 in
denote the weighted average price in period t
t (s)
of the cohort of …rms that last experienced a -shock in period t
t (s)
where
k
s, where all prices are
, i.e.,
= (1
)
s 1 X
k
(
? 1 t k;t Pt s;t k )
+
s
(
? 1 t s;t Pt s;t s )
(13)
:
k=0
There are
s
…rms that have not had a chance to optimally reset prices since receiving
the -shock and (1
)
k
…rms that have last adjusted k < s periods ago. From equation
(6) it follows that one can use the cohort average prices price level as Pt1
=
1 X
(1
)s
t (s)
to express the aggregate
(14)
t (s);
s=0
where
is the mass of …rms that experience a -shock each period and (1
)s is the
share of those …rms that have not undergone another -shock for s periods. To express the evolution of Pt in a recursive form, consider the optimal price Pt?
s;t
of
a …rm that sustained a -shock s > 0 periods ago, but can adjust the price in t due to ? the occurrence of a Calvo shock. Also, consider the price Pt;t of a …rm where a -shock
occurs in period t. The optimal price setting equation (10) then implies ? Pt;t = Pt?
s;t
gt qt
gt qt
s+1
:
(15)
s+1
The previous equation shows that a stronger cohort productivity trend (higher values for q) causes the …rm that experiences a -shock in period t to choose lower prices relative to …rms that experienced -shocks further in the past, as a stronger cohort trend makes this …rm relatively more productive. Conversely, the experience e¤ect (higher values for g) increases the optimal relative price of the …rm that underwent a -shock in t. The net e¤ect depends on the relative strength of the cohort versus the experience e¤ect. 15
Appendix A.4 shows how to combine equations (13), (14), and (15) to obtain a recursive representation for the evolution of the aggregate price level given by Pt1
? 1 = (Pt;t )
+ (1
)(1
)
(pet ) 1
1 ? 1 (Pt;t )
+ (1
)(
1 t 1;t Pt 1 )
;
(16)
where pet summarizes the history of shocks to cohort and experience productivity and evolves recursively according to (pet )
1
) pet 1 gt =qt
= + (1
1
(17)
:
The last term on the r.h.s. of equation (16) captures the price-level e¤ects from the share (1
) of …rms that experienced neither a Calvo shock nor a -shock. These …rms
keep their old price (Pt
1
on average), adjusted for possible e¤ects of price indexation,
as captured by the indexation term
t 1;t .
captures the price e¤ects of the mass
The …rst term on the r.h.s. of equation (16)
of …rms that experienced a -shock in period t;
? . The second term captures the average price of these …rms optimally charge price Pt;t
…rms that experienced a Calvo shock in period t; their share is (1
)(1
) and they set
a price that on average di¤ers from the price charged by …rms hit by a -shock, depending on the value of pet . This latter aspect in equation (16) is the key di¤erence relative to the standard model without …rm heterogeneity in productivity. A stronger experience trend 1
(a higher value for gt ), for instance, increases (pet )
, and - ceteris paribus - causes …rms
hit by a Calvo shock to choose a lower value for the optimal reset price. A stronger cohort trend (a higher value for qt ) has the opposite e¤ect. Overall, the interesting new feature is that price dynamics now depend on the productivity trends. In a setting where all …rms have identical productivity, e.g., where the cohort e¤ect is as strong as the experience e¤ect (qt = gt for all t), equation (17) implies that pet converges to one, causing the price level to eventually evolve according to Pt1
= [ + (1
? 1 )] (Pt;t )
)(1
+ (1
)(
1 t 1;t Pt 1 )
;
which is independent of productivity developments at the …rm level. If in addition there are no -shocks ( = 0), the previous equation simpli…es further to Pt1
= (1
? 1 )(Pt;t )
+ (
1 t 1;t Pt 1 )
;
which describes the evolution of the aggregate price level in the standard Calvo model with homogeneous …rms. Aggregate production function. In appendix A.6 we show that aggregate output Yt can be written as Yt =
At Qt t
1
Kt
16
1
1
Lt
Ft ;
(18)
where Kt denotes the aggregate capital stock, Lt aggregate hours worked and =
t
Z
1
Qt Gjt Qt
0
Pjt Pt
sjt
dj
(19)
evolves recursively according to t
=
+ (1
)(1
)
(pet ) 1
? Pt;t Pt
1
+ (1
)
qt gt
t
t 1:
(20)
t 1;t
TFP in the aggregate production function (18) is a function of the TFP of the latest cohort hit by the -shock, At Qt , and of the adjustment factor
t.
The latter is de…ned
in equation (19) and captures a …rm’s productivity relative to that of the latest cohort, Qt = Qt
sjt Gjt
, and weights this relative productivity with the …rm’s production share
(Pjt =Pt ) . Equations (18) and (19) thus show how relative price distortions may lead to aggregate output losses by negatively a¤ecting aggregate technology, e.g., by allocating more demand to relatively ine¢ cient …rms. The evolution of the adjustment factor over time is described by equation (20) and depends on …rm-level productivity trends - amongst other ways - through the variable pet . In the limit with homogeneous …rm trends (i.e., qt = gt ), pet converges to one and the evolution of
t
becomes independent of productivity
realizations. If - in addition - there are no -shocks ( = 0), then equation (20) simpli…es further to t
= (1
? Pt;t Pt
)
t
+
t 1;
t 1;t
the equation which captures the potential distortions from price dispersion within the standard homogeneous-…rm model. Balanced Growth Path. One can obtain stationary aggregate variables by rescaling them by the aggregate growth trend e t
where
e t
= (At Qt =
e t)
(21)
;
denotes the e¢ cient adjustment factor chosen by the planner, de…ned in equation
(25) below. Speci…cally, the rescaled output yt = Yt = k t = Kt =
e t
e t
and the rescaled capital stock
are now stationary and the aggregate production function (18) can be written
as yt =
e t t
1
kt
1
1
Lt
f
(22)
:
In the deterministic balanced growth path, the (gross) trend growth rate
e t
=
e e t= t 1
is constant and equal to (aq) and hours worked are constant whenever monetary policy implements a constant in‡ation rate. Appendices A.7 and A.8 write all model equations using stationary variables only and appendix A.9 determines the resulting deterministic steady state. 17
5
E¢ ciency of the Flexible-Price Equilibrium
This section derives the e¢ cient allocation and provides conditions under which the ‡exible-price equilibrium is e¢ cient. Appendix B shows that the e¢ cient consumption, hours and capital allocation fCt ; Lt ; Kt+1 g1 t=0 solves max
fCt ;Lt ;Kt+1 g1 t=0
E0
1 X
t
t=0
s:t: Ct + Kt+1 = (1 where
d)Kt +
Z
e t
t
[Ct V (Lt )]1 1
1
0
Qt Gjt Qt
At Qt
1
!
(Kt )1
e t
1
!11
1
dj sjt
(23) (Lt )
1
(24)
Ft ;
(25)
;
which evolves according to (
e 1 t)
= + (1
)
1 e t 1 qt =gt
(26)
:
Constraint (24) is the economy’s resource constraint, when expressing aggregate output using the aggregate production function (18). The e¢ cient productivity adjustment factor
e t
showing up in the planner’s production function is de…ned in equation (25); its
recursive evolution is described by equation (26). The …rst-order conditions of problem (23)-(24) shown in appendix B are necessary and su¢ cient conditions characterizing the e¢ cient allocation. Decentralizing the e¢ cient allocation requires that …rms’prices, which enter
t
and
thus in the aggregate production function (18), satisfy certain conditions. In particular, equation (19) implies that
t
=
e t
is achieved if prices satisfy
Pjt Qt 1 = e Pt t Gjt Qt
(27)
: sjt
The previous equation requires relative prices to accurately re‡ect relative productivities. Furthermore, as in models without …rm heterogeneity, one has to eliminate …rms’ monopoly power by a Pigouvian subsidy to obtain e¢ ciency of market allocation. We thus impose the following condition: Condition 1 The sales subsidy corrects …rms’market power, i.e.,
1 1 1+
= 1.
Appendix C then proves the following result: Proposition 1 The ‡exible-price equilibrium ( = 0) is e¢ cient if condition 1 holds. The proof of the proposition shows that condition (27) holds under ‡exible prices, so that one achieves
t
=
e t
and thereby productive e¢ ciency. In the presence of the
assumed sales subsidy, consumer decisions are also undistorted, which means the values of consumption, hours and capital in the ‡exible-price equilibrium are identical to the values that these variables assume in the e¢ cient allocation. 18
6
Optimal In‡ation with Sticky Prices
This section determines the optimal in‡ation rate for an economy with sticky prices (
> 0). It derives the optimal rate of in‡ation for the nonlinear stochastic economy
with heterogeneous …rms in closed form and shows how in‡ation optimally depends on the productivity growth rates at ; qt and gt . As it turns out, the optimal in‡ation rate implements the e¢ cient allocation (the ‡exible-price benchmark). To establish our main result in the most straightforward manner, we impose an assumption on initial conditions, in particular on how …rms’initial prices and initial productivities are related. Similar conditions are imposed in sticky-price models with homogeneous …rms, where it is routinely assumed that initial dispersion of prices has reached its stationary outcome. We impose: Condition 2 Initial prices in t = Pj;
1
/
1 re‡ect …rms’relative productivities, i.e., 1
Q
1 sj;
1
Gj;
for all j 2 [0; 1]:
1
We discuss the e¤ects of relaxing this condition below. The following proposition states our main result: Proposition 2 Suppose conditions 1 and 2 hold. The equilibrium allocation in the stickyprice economy is e¢ cient if monetary policy implements the gross in‡ation rate ? t
where
? t 1;t
=
? t 1;t
1
(
1
e t)
1
!
1 1
for all t
captures price indexation between periods t 1 and t (
of indexation) and
e t
(28)
0; ? t 1;t
1 in the absence
is de…ned in equation (25) and evolves according to equation (26).
In the absence of price indexation ( a function of the variable
e t,
? t 1;t
1), the optimal in‡ation rate
? t
is only
which captures the distribution of relative productivities
between all …rms …rms and those with a -shock; see equation (25). Since these relative productivities are independent of the common TFP growth rate at , it follows that the optimal in‡ation rate does not depend on the realizations of at . In contrast, the cohort productivity growth rate qt and the experience growth rate gt do a¤ect
e t,
see equation
(26). Yet, these trends a¤ect the optimal in‡ation rate in opposite directions: a stronger cohort productivity growth rate qt decreases the optimal in‡ation rate, while a stronger experience growth rate gt increases the optimal in‡ation rate. For the special case in which all …rms have identical productivity trends ( = 0 or gt = qt ) or even identical productivities (
e t
= 1), the optimal gross in‡ation rate is equal 19
to one in the absence of price indexation, as in a standard homogeneous-…rm model. Perfect price stability is then optimal at all times. Price indexation by non-adjusting …rms (
? t 1;t
6= 1), say because of indexation to
the lagged in‡ation rate, introduces additional components into the optimal aggregate
in‡ation rate. In particular, it requires that price-adjusting …rms, i.e., …rms hit by either a -shock or a Calvo shock, also adjust their price by the indexation component. This way prices continue to accurately re‡ect relative productivities at all times. This explains why indexation a¤ects the optimal in‡ation rate one-for-one. Although proposition 2 assumes that …rms’initial prices accurately re‡ect the initial relative productivities, the initial productivity distribution itself is unrestricted. We conjecture that for a setting where condition 2 fails to hold, one would obtain additional transitory and deterministic components to the optimal in‡ation rate, as in the homogeneous …rm setting studied by Yun (2005). The in‡ation rate stated in proposition 2 would then become optimal only asymptotically. The proof of proposition 2, which is contained in appendix D, establishes that with the optimal in‡ation rate …rms choose relative prices as in the ‡exible-price equilibrium. This result is established by showing that (1) …rms hit by a -shock choose the same optimal relative price as in the ‡exible price economy, and that (2) …rms hit by a Calvo shock optimally choose not to adjust their price, which avoids the emergence of price dispersion between otherwise identical …rms. This, together with the fact that (3) initial prices re‡ect initial productivities, ensures that all relative prices are identical to those in the ‡exible-price equilibrium. Under the assumed output subsidy, it then follows that household allocations are also identical to the ‡exible-price equilibrium, which has been shown to be e¢ cient; see proposition 1. Interestingly, it follows from the proof of proposition 2 that the in‡ation rate (28) continues to ensure productive e¢ ciency (but not full e¢ ciency) in settings where condition 1 fails to hold. From the theory of optimal taxation it then follows that it remains optimal to implement the in‡ation rate (28), as it is suboptimal to distort intermediate production as long as (distortionary) taxes on …nal goods are available.
7
The Optimal Steady-State In‡ation Rate
This section discusses the optimal steady-state in‡ation rate implied by the model. To simplify the discussion, we abstract from price indexation, unless otherwise stated. Proposition 2 makes it clear that in the case in which the productivity of all …rms grows at the same rate ( = 0), which includes as a special case a setting with homogeneous …rms, we obtain that the optimal in‡ation rate For
? t
= 1, independently of all shock processes.
= 0, the optimal (gross) steady-state in‡ation rate is thus trivially equal to one.
20
For the case from
? t
> 0, the optimal steady-state in‡ation rate jumps discontinuously away
= 1, but turns out to be itself independent of the value of .18 The following
lemma summarizes this result: Lemma 1 Suppose conditions 1 and 2 hold, there are no economic disturbances, there is no price indexation (
? t 1;t
1) and
> 0. The optimal in‡ation rate then satis…es lim
t!1
? t
(29)
= g=q:
Proof. From equations (7) and (26) it follows that ( then follows from proposition 2 that limt!1
? t
e t)
= g=q.
1
! [1
(1
) (g=q)
1
]= . It
Since we allow for arbitrary initial productivity distributions, the absence of shocks does not necessarily imply that the optimal in‡ation rate is constant from the beginning. This only happens asymptotically, once the productivity distribution converges to its stationary distribution (in detrended terms).19 The lemma provides the in‡ation rate that is asymptotically optimal as this stationary distribution is reached.20 Interestingly, the optimal long-run in‡ation rate is completely independent of the intensity of -shocks, which may appear surprising. To understand the source of this invariance, consider a setting where -shocks capture …rm exit and entry events and where g > q, so that entering …rms are smaller and less productive than the set of non-exiting …rms. A higher value for
implies that more young and relatively unproductive …rms are
amongst the set of price-setting …rms. This calls - ceteris paribus - for a higher in‡ation rate. Yet, the productivity distribution of non-exiting …rms is not invariant to changes in : a higher
also implies more …rm turnover and thus less experience accumulation.
Non-exiting …rms thus tend to be less productive relative to new entrants, which calls for lower in‡ation rates. In net terms, these two e¤ects exactly cancel each other. On empirical grounds, it appears plausible to assume g > q, so that according to lemma 1 the optimal steady-state in‡ation rate is positive. For the case with …rm turnover, the fact that young …rms are small relative to old …rms requires g > q. Likewise, interpreting -shocks as representing product substitution shocks, the case g > q implies that new products are relatively more expensive than old products and that their relative price is falling over the life cycle of the product. Both of these facts are in line with evidence 18
Note that the e¢ cient allocation also discontinuously jumps when moving from = 0 to > 0, as in the former case e¢ cient aggregate growth is equal to (ag) and in the latter case it is equal to (aq) in steady state. Appendix E shows that the discontinuity of the optimal steady-state in‡ation rate is not due to the discontinuity of the e¢ cient real allocation. 19 This is not an issue when = 0: the initial distribution then remains unchanged (in detrended terms). 20 The transitional dynamics can easily be derived from proposition 2 using the initial productivity distribution and equation (26).
21
provided by Melser and Syed (2016) but would not be obtained if we assumed g < q. Thus, while the setup allows the optimal steady-state in‡ation rate to be potentially negative, these considerations suggest positive in‡ation to be optimal in steady state. Interestingly, aggregate productivity dynamics turn out not to be informative about the optimal in‡ation rate. The aggregate steady-state growth is equal to (aq) and is driven by a factor that a¤ects the optimal in‡ation rate (q) and a factor that does not a¤ect it (a). Moreover, the experience e¤ect (g) has no aggregate growth rate implications, but a¤ects the optimal in‡ation rate. Determining the optimal in‡ation rate thus requires studying the …rm-level productivity trends g and q, as aggregate productivity fails to identify the in‡ation-relevant productivity trends. We shall come back to this issue in our empirical section 11. Finally, we discuss the e¤ects of price indexation. For in‡ation rate is then given by in‡ation according to
? t 1;t
=
? t 1;t (g=q). For the ? for some t 1
lim
t!1
? t
> 0 the optimal long-run
case where prices are indexed to lagged 2 [0; 1), we obtain
= (g=q) 1
1
:
Standard forms of price indexation thus amplify the divergence of the optimal gross in‡ation rate from one.
8
The Welfare Costs of Strict Price Stability
This section shows that suboptimally implementing strict price stability, as suggested by sticky-price models with homogeneous …rms, gives rise to strictly positive welfare costs whenever g 6= q. We derive this fact …rst analytically for a special case. The analytic result allows us to consider also the limit
! 0. In a second step, we use numerical
simulations to highlight the source of the welfare losses and their magnitude.
The following proposition shows that - as long as g 6= q - there is a strictly positive
welfare loss that is bounded away from zero when implementing strict price stability; this holds true even for the limit
! 0.21
Proposition 3 Suppose conditions 1 and 2 hold, there are no economic disturbances, > 0, …xed costs of production are zero (f = 0), there is no price indexation (
? t 1;t
1),
and the disutility of work is given by V (L) = 1 with
> 1 and
> 0. Assume g=q > (1
L ; ), so that a well-de…ned steady state with
strict price stability exists. 21
The proof of the proposition is contained in appendix F.
22
A. Relative cohort price: cohort mean, different inflation rates
1.2
B. Relative cohort price: mean and 2 std.dev. bands 0% inflation rate.
1.2
1.15
1.15
1.1 1.1 1.05 1.05
0% Inflation Rate Optimal Inflation Rate: 2%
1
1
0.95 0
5
10 15 cohort (time since last
20 -shock)
25
0
5
10
15
20
cohort (time since last
25
-shock)
Figure 2: Relative prices and in‡ation
Consider the limit ( e )1 ? t
! 1 and a policy implementing the optimal in‡ation rate
from proposition 2, which satis…es limt!1
? t
=
?
= g=q. Let c(
?
) and L(
?
)
denote the limit outcomes for t ! 1 for consumption and hours, respectively, under this
policy. Similarly, let c(1) and L(1) denote the limit outcomes under the alternative policy of implementing strict price stability. Then, L(1) = L(
?
)
and c(1) = c( ? )
1
(1 1
)(g=q) (1 )
1
1
1 1
For g 6= q the previous inequality is strict and lim
(1 (1 !0
) (g=q) )(g=q)
c(1)=c(
?
1 1
!
1:
(30)
) < 1:
We now illustrate the nature of the relative price distortions that are generated by suboptimal rate of in‡ation and how they give rise to welfare losses. Panel A in …gure 2 reports the mean cohort price (relative to the price of all …rms), depicted on the yaxis, as a function of the cohort age in quarters (x-axis). It does so once for a setting where monetary policy implements a 2% in‡ation rate annually (in net terms) and once when monetary policy pursues strict price stability. The assumed optimal in‡ation rate is thereby 2%.22 Panel A shows that young cohorts charge a higher (relative) price and that this price decreases over the lifetime of the cohort. Under the optimal in‡ation rate (2%) the decline happens at a constant rate.23 Under strict price stability, …rms anticipate that their relative prices will not necessarily fall, due to Calvo price stickiness. This causes them to initially "front load" prices, i.e., in an environment with strict price stability 22
Figure 2 is computed using g = 1:020:25 ; q = 1; = 0:75; = 0:035; = 3:8 and assumes that the initial productivity distribution is equal to the stationary distribution (in detrended terms). 23 The …gure assumes that no shocks hit the economy.
23
young cohorts charge initially lower prices than under the optimal in‡ation rate. Over time, some …rms in the cohort will get the opportunity to lower their prices in response to Calvo shocks, but the average relative price of the cohort will eventually be slightly higher than under the optimal in‡ation rate. Beyond these distortions in average cohort prices, the suboptimal in‡ation rate also generates prices distortions within a cohort of …rms. This is illustrated in panel B of …gure 2. Panel B reports the mean cohort price and the +/-2 standard deviation bands of the cross-sectional price distribution within the cohort, assuming monetary policy pursues strict price stability. It shows that suboptimal in‡ation not only gives rise to distortion in mean prices but also to substantial amounts of price dispersion within the cohort. Under the optimal in‡ation rate, price dispersion is zero at the cohort level. e t=
Figure 3 reports the steady-state value for the ratio
t
(y-axis) as a function of
the implemented steady-state in‡ation rate (x-axis), when the optimal in‡ation rate is 2% per year.24 The aggregate production function (18) shows that one can interpret
e t=
t
as a measure of the aggregate productivity distortion that is implied by the relative price distortions associated with suboptimal in‡ation rates.25 The …gure shows that a 10% shortfall of the in‡ation rate below its optimal value of 2% is associated with an aggregate productivity loss equal to about 1%. In the process, the productivity losses arise rather nonlinearly: a shortfall of in‡ation of 2% below its optimal value is associated with an aggregate productivity loss of just 0.05%. Furthermore, in‡ation losses are asymmetric, with above-optimal in‡ation leading to relatively larger losses. For instance, increasing in‡ation 8% above its optimal value generates a productivity loss of 0.94%, while decreasing in‡ation by the same amount below its optimal value leads to a productivity losses of only 0.37%.
9
The Variance of the Optimal In‡ation Rate
This section discusses the optimal dynamic response of in‡ation to productivity disturbances. In the absence of -shocks, we obtain from proposition 2 ? t
= 1 for all t;
i.e., the optimal in‡ation rate is then independent of productivity shocks at all times. For
> 0, it follows from equations (26) and (28) that the optimal nonlinear in‡ation
response to productivity disturbances is given by 1 1 24 25
(1
)
? t t 1;t
1
(1
)
(1
)
=1+ 1
The …gure is based on the same parameterization as …gure 2. Appendix F shows that c(1)=c( ? ) = ( e = ) :
24
1
gt qt ? t 1
t 2;t 1
1
:
(31)
1.01
Aggregate productivity distortion
e
/
1.005
1
0.995
0.99
0.985
0.98 -10
-8
-6
-4
-2
0
2
4
6
8
Steady state inflation rate (annualized, in %)
Figure 3: Aggregate productivity as a function of steady state in‡ation (optimal in‡ation rate is 2%)
Abstracting from price indexation ( ? t
As long as
= (1
t 1;t
)
1), a linearization of equation (31) delivers26 ? t 1
gt qt
+
(32)
1 :
< 1, the optimal in‡ation rate will thus display persistent responses to any
deviation of gt =qt from its average value. A positive surprise to experience productivity growth gt , for example, shifts up permanently the experience level of old cohorts. This requires persistently higher in‡ation rates, as new cohorts (…rms hit by a -shock) now have to keep raising their prices continuously until the productivity distribution returns to its stationary distribution (in detrended terms). The speed with which the productivity distribution returns to its stationary distribution depends on . For
= 1, the return
is immediate and the optimal in‡ation rate inherits the persistence properties of the exogenous driving process gt =qt . For values of
close to zero, the optimal in‡ation rate
approximately behaves like a random walk whenever gt =qt is an i.i.d. process, but the unconditional variance of in‡ation decreases with 26
We log-linearize with respect to the variables
? t
25
and approaches zero as
and gt =qt at the point (
? t ; gt =qt )
! 0.
= (1; 1):
10
Extension to a Multi-Sector Economy
We now extend the basic sticky-price setup to a multi-sector setting that allows (inter alia) for sector-speci…c productivity trends and sector-speci…c price stickiness. Such an extension is relevant when seeking to bring the model to the data, as we do in the next section, because productivity trends and price stickiness tend to be di¤erent across the manufacturing and the service sectors. We show below that the optimal steady-state in‡ation rate in a multi-sector economy is a weighted average of the in‡ation rates that would achieve e¢ ciency in the respective sectors individually. Consider an economy with z = 1; : : : ; Z sectors in which aggregate output Yt is Yt =
Z Y
(Yzt )
z
;
z=1
with Yzt denoting output in sector z and z P with the expenditure shares satisfying Zz=1
0 being the sector’s expenditure share, z
= 1. Sectoral output Yzt itself is a Dixit-
Stiglitz aggregate of the output of a unit mass of …rms j in sector z, in close analogy to the one-sector setup. Let Azt denote the TFP component, Qzt the cohort-speci…c component and Gjzt the experience component to …rm productivity in sector z = 1; : : : ; Z. Output of the …rm producing product j 2 [0; 1] in sector z is then given by Yjzt = Azt Qt
1
1
1
sjzt Gjzt Kjzt Ljzt
Fzt ;
where sjzt denotes the number of periods since the last -shock, Kjzt employed capital, Ljzt employed labor, and Fzt
0 a sector-speci…c …xed cost of production. The sector-speci…c
growth rates of Azt ; Qzt and Gjzt are given by azt = az
a zt ,
qzt = qz
q zt
and gzt = gz
g zt ,
respectively, where az , qz and gz denote the steady-state growth trends. We also allow for sector-speci…c degrees of price stickiness intensities
z
z
2 (0; 1) and for sector-speci…c -shock
2 (0; 1). We do not introduce additional sectors speci…c frictions, i.e., we
assume the existence of a common capital and labor market. The aggregate price level is de…ned as Pt =
Z Y z=1
Pzt
z
;
z
where the sectoral price level Pzt depends on the prices charged by …rms in sector z in the same way as in the one-sector economy; see equation (6). Further details of the multisector economy are provided in a separate technical appendix, jointly with the proof of the following proposition.
26
Proposition 4 Suppose condition 1 holds, there are no economic disturbances, there is no price indexation (
? t 1;t
1), there is positive -shock intensity in all sectors ( e 1
z
>0 e
for all z = 1; : : : Z), and the discount factor approaches ( ) ! 1, where = QZ z denotes the growth trend of the aggregate economy. Suppose monetary z=1 (az qz ) policy implements the in‡ation rate
for all t. The in‡ation rate
=
t
that maximizes
utility in the steady state of the multi-sector economy is ?
=
Z X
gz qz
!z
z=1
where
e z e
= QZ
e z e
(33)
;
az q z
z=1 (az qz )
z
denotes the growth trend of sector z relative to the growth trend of the aggregate economy. The sectoral weights !z
0 are given by ! ~z !z = PZ
~z z=1 !
with ! ~z =
[1
z (1
z )(
e=
z z (1 e) ( ?) z
= ze ) ( (qz =gz )] [1 z )(
e
;
?
) (qz =gz ) e e z (1 z )( = z )
1(
?)
1]
:
The proposition shows that the result from the one-sector economy naturally extends to a multi-sector setup. The main new element consists of the fact that the sector-speci…c optimal in‡ation rates gz =qz need to be rescaled by the sectors’ relative growth trends e e z= .
This implies that the sector-speci…c optimal in‡ation rate gz =qz is scaled upwards
for faster growing sectors ( ze =
e
> 1) and scaled downwards for sectors that grow slower
than the aggregate economy. Since proposition 4 provides a result specifying the in‡ation rate that maximizes utility in the limiting steady state, rather than a result about the limit of the optimal in‡ation rate itself, we do not have to impose condition 2, unlike in proposition 2. Furthermore, unlike in proposition 1, the optimal in‡ation rate fails to implement the …rst-best allocation, which would generally require di¤erent in‡ation rates for di¤erent sectors. The limiting condition
( e )1
! 1 is required in proposition 4 to ensure that the utility
losses due to aggregate markup distortions and those due to relative price distortions are
minimized by the same in‡ation rate, namely the one given in the proposition. Absent this condition, minimizing these distortions individually would call for di¤erent in‡ation rates. One could then use sector-speci…c output subsidies to undo the sectoral markup distortions. The in‡ation rate
?
stated in proposition 4 is then optimal even if ( e )1
is strictly smaller than unity. 27
For the special case with az qz = aq, which implies that obtain from equation (33) that
?
e z
=
e
, and gz =qz = g=q, we
= g=q, which is the result for the one-sector economy
stated in lemma 1. In this special case, the central bank does not face a trade-o¤ between di¤erent sector-speci…c optimal in‡ation rates and can achieve the …rst-best allocation in the resulting steady state of the multi-sector economy, despite the presence of sectorspeci…c degrees of price stickiness and sector-speci…c -shock intensities. Since the closed-form expressions for the sector weights !z in proposition 4 are di¢ cult to interpret and also depend on the optimal in‡ation rate, the subsequent lemma shows that these weights are - to a …rst order approximation - equal to the sector’s expenditure weights
z:
Lemma 2 The optimal steady-state in‡ation rate in the multi-sector economy is equal to =
Z X
z
z=1
gz qz
e z e
(34)
+ O(2);
where O(2) denotes a second order approximation error and where the approximation to equation (33) has been taken around a point, in which
gz qz
e z e
and
z (1
z )(
e
= ze )
1
are
constant across sectors z = 1; : : : Z.
Interestingly, the optimal steady-state in‡ation rate turns out to be (to …rst order) independent of the sector-speci…c degree of price stickiness (
z ),
unlike in Benigno (2004).
This happens because the point of approximation is chosen such that sector-speci…c productivity trends and the e¤ects of sector-speci…c price stickiness cancel each other. Our result then shows that to a …rst order approximation, the optimal in‡ation rate remains independent of
z
in the neighborhood of this point. This contrasts with the e¤ects of
sector-speci…c …rm-level productivity trends (qz =gz ), which do have …rst order implications for the optimal steady-state in‡ation rate.
11
The Optimal In‡ation Rate for the US Economy
This section quanti…es the optimal in‡ation rate for the US economy using the multisector setup presented in the previous section. The next section explains our empirical approach and the subsequent section presents the estimation results.
11.1
Empirical Strategy
To quantify the optimal in‡ation rate implied by microeconomic productivity trends, one would ideally estimate these trends directly at the …rm or establishment level. Yet, it is
28
generally di¢ cult to measure physical productivity at the …rm or establishment level because output prices are not widely observed at this level of observation.27 Given this, we proceed by using establishment-level employment trends to estimate the establishmentlevel productivity trends. Employment and productivity are related to each other via the elasticity of product demand (
1), which maps any …rm-level productivity (and associ-
ated product price) di¤erence into an employment di¤erence.28 Clearly, to the extent that …rms face additional constraints for expanding production beyond being insu¢ ciently productive/competitive (e.g., …nancial constraints, adjustment costs, regulatory constraints), there may be biases in the productivity trends that are estimated from employment trends. To deal with this concern, we shall mainly look at changes in the estimated trends over time, which should remove any …xed e¤ects that arise from other frictions a¤ecting …rmlevel employment. We estimate the employment trends using information provided by the Longitudinal Business Database (LBD) of the US Census Bureau. The LBD reports establishmentlevel employment data and covers all US establishments at an annual frequency. Coverage starts in the year 1976 and we use data up the year 2013. For this period, there are a total of 176 million employment observations at the establishment level. Working with this data source, we shall interpret -shocks as an event in which the establishment is closed down, in line with the "…rm entry and exit" interpretation spelled out in section 3.2. The variable sjzt can then be interpreted as the establishment age in sector z. Using the multi-sector setup from the previous section, we then can derive a model-implied relationship between establishment-level employment, establishment age, both of which are observed in the LBD, and the productivity trends of interest:29 27
As explained in Foster, Haltiwanger and Syverson (2008), the productivity literature usually measures revenue productivity instead of physical productivity at the …rm level, which de‡ates …rm-level output with some industry-level price index. In our setting, …rms’revenue productivity is completely unrelated to their physical productivity in the absence of …xed costs of production. For the few industries for which physical and revenue productivities can both be observed, the two productivity measures can be rather di¤erent; see Foster, Haltiwanger and Syverson (2008). 28 An alternative approach would consist in considering product-level price data, e.g., the price information entering into the construction of the CPI, as used for example in Nakamura and Steinsson (2008). The results documented in Bils (2009) and Moulton and Moses (1997) show that the in‡ation rate of so-called "forced substitution" items, i.e., items which become permanently unavailable and are replaced by other "new" items, is signi…cantly larger than that of so-called matched items, which are products that continue to be available. Our model implies that one can infer the in‡ation-relevant trend g=q from the in‡ation di¤erence in these two-item categories. However, this would require making accurate quality adjustments in the computation of the in‡ation rate for forced substitution items, which is a task that is di¢ cult to achieve. 29 The proof of the following proposition can be found in a separate technical appendix which also spells out the details of the multi-sector setup.
29
Proposition 5 Suppose that
Pt
i=0
ln(
q g zi = zi )
is a stationary process in t and that …xed
costs are equal to zero (fz = 0). Employment Ljzt of the …rm producing product j in sector z at time t in the ‡exible-price equilibrium is then equal to ln(Ljzt ) = dzt +
z
sjzt +
(35)
jzt ;
where dzt denotes a sector-speci…c time dummy, sjzt the age of the …rm and ary residual term. The regression coe¢ cient z
The requirement that
Pt
i=0
arity of the regression residuals
ln(
=( q g zi = zi )
jzt .
z
jzt
a station-
is given by
1) ln(gz =qz ): is stationary is essential for obtaining station-
It is satis…ed, for instance, if ln Qzt and ln Gjzt are
both trend stationary or non-stationary but co-integrated processes. The coe¢ cient of interest,
z,
captures the in‡ation-relevant productivity trends, multiplied by elasticity of
product demand (
1).
Note that proposition 5 derives a property about …rm-level employment in the ‡exibleprice equilibrium. In the one-sector economy, the optimal in‡ation rate replicates the ‡exible-price equilibrium exactly, indicating that the ‡exible-price employment would indeed be observed in equilibrium under optimal monetary policy. This fails to be exactly true in the multi-sector economy or when policy is conducted in a suboptimal way. Relative price distortions associated with sticky prices then generally a¤ect the equilibrium distribution of …rm-level employment. Since these distortions do not tend to systematically vary with …rm age, they will likely get absorbed by the …rm-level residuals in equation (35). Moreover, since we identify trends by looking at yearly observations, the more short-lived e¤ects of price stickiness at the …rm level can plausibly be expected to be averaged out. Note also that equation (35) holds whenever …xed costs of production are zero. From the proof of proposition 5 it follows that further nonlinear terms in age can show up on the right-hand side of equation (35), for strictly positive …xed costs. In our empirical analysis, we therefore explore the robustness of the estimates
z
when including also age
squared as a regressor on the right-hand side. Equation (35) is of interest because it allows us to identify, when combined with information about the demand elasticity , the sector-speci…c relative productivity trends gz =qz from establishment-level data. The values for gz =qz for all sectors together then determine - jointly with the sector speci…c relative growth trends about sector size
z
e e z=
and information
- the optimal aggregate in‡ation rate, as implied by lemma 2.
To avoid censoring of the age variable when estimating
z,
we consider a restricted
LBD sample that excludes the …rms that are already present in the initial year (1976), for which age information is not available. Furthermore, we start estimating 30
z
from
the year 1986 onwards, so as to minimize any e¤ects from having only young establishments in the sample during the initial years of the database. This leaves us with 147 million establishment-age observations. The mean age for this sample is 8.16 years, with a standard deviation of 7.05 years, which means that a wide range of age observations is covered. We then estimate equation (35) for 65 private BEA industries, which is the level of disaggregation at which sectoral GDP information is available. The sectoral GDP information allows us to compute the sector-speci…c relative growth trends
e e z=
showing
up in equation (34). To this end, we map the NAICS industries in the LBD database into the 65 private BEA industries (for the early part of the sample we map SIC codes). Finally, the GDP weights
z
are computed using sectoral GDP information for the year
2013. As another robustness exercise, we use time-varying weights
zt
using sectoral GDP
information for each of the considered years t = 1986; : : : 2013. Details of our approach are provided in appendix G.
11.2
Empirical Results
For the years 1986 to 2013, we report the time series t
65 X
e z e
z
(36)
exp(bzt );
z=1
where bzt is the time t OLS estimate for
z,
for sector z = 1; : : : 65. The time series
t
is
30
of interest, because it is proportional to the optimal steady-state in‡ation rate ? t
where forming
? t
1=
1 1
(
t
1) + O(2);
(37)
is the time t estimate of the steady-state in‡ation rate from lemma 2. Transt
into an implied in‡ation rate thus requires us to take a stand on the value
of the demand elasticity . For statements about the relative evolution of the optimal in‡ation rate, the value of
does not matter. For the year 2007, which is the last year
before the start of the …nancial crisis, we report in appendix G.3 detailed information on the cross-sectional estimates bz , descriptive statistics for the various sectors, and the
outcome of a robustness exercise. For the year 2007, we can also compute the estimation uncertainty about
t,
which turns out to be very small: the standard error is below one
basis point.31 Figure 4 presents our baseline estimate for
t.
It shows that the optimal in‡ation
rate is positive, in line with the empirical observation that older establishments tend to e P This follows from exp(( 1) ln(gz =qz )) = 1 + ( 1)(gz =qz 1) + O(2) and z z ze = 1 + O(2). 31 We use the delta method to compute the uncertainty about t assuming that the individual estimates bz are independent across z. The latter assumption is needed as we do not have information about the full covariance matrix for the vector consisting of the elements z , z = 1; :::; 65.
30
31
1.07
1.065
1.06
1.055
1.05
1.045
1.04
1.035
1.03
1.025 1990
1995
2000
2005
2010
Year
Figure 4: Baseline estimate of
t
(…xed 2013 sector weights, linear speci…cation in age)
employ more workers on average (gz =qz > 1). It also shows that the optimal in‡ation rate dropped by approximately …fty percent over the period 1986 to 2013. The decline is rather steady over time and there are only weak indications for cyclical ‡uctuations. The drop in the estimate implies that either the experience trend in productivity weakened over the considered time period or the cohort productivity trend a¤ecting new entrants accelerated.32 While we cannot disclose the cross-sectional estimates for the year 2002, comparing 2002 estimates of
z
to the 2007 estimates reported in appendix G, we …nd
that the decline in bz is widespread in most economic sectors and not driven by a small set of sectors experiencing very large declines.
Figure 5 investigates the robustness of the baseline estimates to alternative estima-
tion approaches. As a …rst alternative, we consider weights
zt
that re‡ect the sectoral
GDP share of each economic sector in period t rather than using …xed weights from the year 2013. This is motivated by the observation that the GDP shares of sectors have shifted considerably over the considered period. For instance, the share of manufacturing in private GDP dropped from 21.1 percent in 1986 to 14.0 percent in 2013. As …gure 5 shows, using these time-varying weights leads to only negligible changes in the estimates. The sectoral rebalancing taking place in the US economy thus does not co-vary signi…cantly with the changes in the bz (and thus gz =qz ). This is consistent with the previous observation that the drop in
z
is present in almost all sectors of the US economy.
Figure 5 also presents estimates for
32
It is impossible to identify from the
z
t
when additionally including a term in age
estimates, which of the two e¤ects actually drives the decline.
32
1.08
1.07
1.06
1.05
1.04
1.03
1.02 1990
1995
2000
2005
2010
Year
Figure 5: Robustness of
t
estimate to alternative estimation approaches
squared on the right-hand side of the regression equation (35).33 The
t
estimates then
become slightly larger in magnitude and also considerably more cyclical, displaying drops around the recession years 1991, 2001, and 2009. The overall message, however, remains unchanged: The optimal in‡ation rate is positive and approximately halved over the considered time period. Translating the estimates presented in …gures 4 and 5 into optimal in‡ation rates requires us to take a stand on the value of the demand elasticity parameter . As our baseline, we follow Bilbiie et al. (2012) and Bernard et al. (2003) who use
= 3:8 based
on a calibration that …ts US plant and macro trade data. As a robustness exercise, we also consider
= 5, based on a calibration in Eusepi et al. (2011) that …ts the revenue
34
labor share.
Table 1 reports the outcomes for the optimal in‡ation rate over the sample period. It shows that, for most of the speci…cations, the optimal in‡ation rate dropped from a value of close to two percent in 1986 to approximately one percent in 2013. The optimal steady-state in‡ation rate thus appears to have dropped signi…cantly over these 27 years. 33
The de…nition of t is still given by equation (36). Eusepi et al. (2011) also show that the average wholesale markup implies a demand elasticity of 5.1 for the industries in the 1997 Census of Wholesale Trade that Bils and Klenow (2004) could match to consumer goods in the CPI. 34
33
Table 1: Optimal In‡ation Rate (Net) Baseline TV Weights LQ Speci…cation Baseline TV Weights LQ Speci…cation = 3:8 ? 1986 ? 2013
=5
2.34%
2.24%
2.70%
1.64%
1.57%
1.89%
1.02%
1.02%
1.45%
0.71%
0.71%
1.01%
Notes: "Baseline" refers to the baseline estimate of "TV Weights" refers to the estimate of refers to the estimate of The parameter
12
t
t
t
with …xed GDP weights and age as single regressor.
that is based on time-varying GDP weights. "LQ Speci…cation"
that is based on a speci…cation with both age and age squared as regressors.
denotes the product demand elasticity.
Extensions and Robustness of Results
This section considers various extensions and alternative model setups. Section 12.1 shows that our main …nding about the optimal in‡ation rate (proposition 2) continues to apply in a setting where price adjustment frictions take the form of menu costs. Section 12.2 discusses the e¤ects of introducing additional …rm-speci…c productivity components.
12.1
Menu Cost Frictions
While our results are illustrated using time-dependent Calvo price-setting frictions, our main theoretical …nding from proposition 2 extends to a setting in which …rms have to pay a …xed cost to adjust their price. Since the optimal in‡ation rate in proposition 2 replicates the ‡exible-price allocation, …rms that do not experience a -shock have - independently of the nature of their price setting frictions - no incentives to adjust their prices, whenever monetary policy implements the optimal in‡ation rate. Since the ‡exible price allocation is e¢ cient, see proposition 1, monetary policy also has no incentive to deviate from the ‡exible-price allocation. Both observations together imply that the optimal in‡ation rate does not depend on whether price-setting frictions are state or time dependent.35 The previous logic does not fully extend to our results for a multi-sector economy in section 10. The optimal in‡ation rate there fails to exactly implement the e¢ cient ‡exible-price allocation, because generically each sector has its own sector-speci…c optimal in‡ation rate. The precise form of the postulated price setting frictions can then have an in‡uence on the optimal rate of in‡ation, as it determines the details of how monetary policy can get allocations closer to the e¢ cient ‡exible-price benchmark. Determining 35
Obviously, this requires that, in a setting with menu cost frictions, -shocks lead either to these menu cost not having to be paid or always having to be paid, independently of whether or not prices are adjusted. The latter situation appears plausible when interpreting -shocks as being associated with new products, new product qualities or …rm entry and exit; see section 3.2.
34
the optimal in‡ation rate for menu cost type models, in which monetary policy cannot replicate the e¢ cient allocation, e.g., the setting considered in Golosov and Lucas (2007), is of interest for further research but beyond the scope of the present paper.
12.2
Idiosyncratic Firm Productivity
The model that we present allows for a considerably richer …rm-speci…c productivity process than many other sticky-price models. At the same time, however, it abstracts from a number of potentially interesting additional dimensions of …rm-level heterogeneity. In particular, one simplifying assumption entertained throughout the paper is that there are no …rm-speci…c productivity components: …rms receiving a -shock, for example, are homogeneous and - absent further -shocks - their productivity grows according to common trends de…ned by (at ; gt ). This said, some of our results continue to hold even in the presence of additional idiosyncratic elements to …rm productivity. Adding to the setup, for instance, a multiplicative …rm …xed e¤ect to productivity, i.e., letting …rm-speci…c productivity be given by Zjt = Qt where Zej;t
sjt
e
sjt Gjt Zj;t sjt ;
is chosen in the period in which a -shock hits, independently across …rms
and from a time-invariant distribution with mean one, our main results in proposition 1 and 2 continue to apply. The same holds true if we incorporated instead a time-invariant …rm-…xed e¤ect Zej for productivity, i.e., Zjt = Qt
e
sjt Gjt Zj :
For these extended settings, proposition 5, which derives our model implied empirical speci…cation for estimating the relevant sectoral productivity trends, also holds in unchanged form because …rm …xed e¤ects get absorbed by the …rm speci…c error term. The situation is di¤erent if …rm-speci…c components are time-varying, e.g., take the form Zjt = Qt
e
sjt Gjt Zjt
for some idiosyncratic productivity shock process Zejt with unconditional mean one. While such idiosyncratic shocks may be present in the data, it is hard to know to what extent they represent measurement noise. In any case, the presence of such shocks prevents full
replication of the ‡exible-price equilibrium, as adjustment frictions then become strictly binding.36 While this makes it di¢ cult to obtain closed-form solutions for the optimal in‡ation rate, studying the implications of such time-varying idiosyncratic shocks for the optimal in‡ation rate appears to be worth exploring further in future work. 36
This holds true for the case with time-dependent pricing frictions and the case with menu cost type frictions.
35
13
Conclusions
This paper shows how …rm-level productivity trends a¤ect the in‡ation rate that is optimal for the aggregate economy. Since the in‡ation-relevant …rm-level productivity trends cannot be inferred from aggregate productivity dynamics, we analyze data from the Longitudinal Business Database, which reports establishment-level employment for all US establishments. We …nd that the US employment trends imply establishment-level productivity trends that can rationalize signi…cantly positive rates of in‡ation as being optimal. At the same time, our estimates show that important changes in …rm-level productivity trends have been taking place over the period 1986 to 2013 in the US economy. These changes caused the optimal in‡ation rate to fall by approximately 50 percent over the considered 27 years. The economic forces behind these changes in establishment-level productivity trends are certainly worth exploring further. It also appears interesting to explore to what extent similar changes are present in other advanced economies.
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39
Appendix A
Derivation of the Sticky-Price Economy
A.1
Cost Minimization Problem of Firms
The cost minimization problem of …rm j, min Kjt rt + Ljt Wt =Pt
s:t: Yjt = At Qt
Kjt ;Ljt
1
sjt Gjt
Kjt
1
1
Ljt
Ft ;
yields the …rst-order conditions 1
0 = rt + 1 0 = Wt =Pt + where
t
1
t At Qt sjt Gjt
t At Qt
1
Ljt Kjt
sjt Gjt
1
Ljt Kjt 1
;
denotes the Lagrange multiplier. The …rst-order conditions imply that the
optimal capital labor ratio is the same for all j 2 [0; 1], i.e., Wt Kjt = ( Ljt Pt r t
1):
Plugging the optimal capital labor ratio into the technology of …rm j and solving for the factor inputs yields the factor demand functions Ljt =
Wt ( Pt r t
Kjt =
Wt ( Pt r t
1
1
1)
(38)
Ijt 1
(39)
Ijt :
1)
Firm j demands these amounts of labor and capital, respectively, to combine them to Ijt , which yields Yjt units of output. Accordingly, the …rm’s cost function to produce Ijt is M Ct Ijt = Wt
1
Wt ( Pt rt
1
1)
Ijt + Pt rt
Wt ( Pt r t
1
1)
(40)
Ijt ;
where M Ct denotes nominal marginal (or average) costs. This equation can be rearranged to obtain equation (8) in the main text.
A.2
Price-Setting Problem of Firms
The price-setting problem of the …rm j, see equation (9), implies that the optimal product price is given by Pjt? =
1 11+
Et
P1
i=0 (
Et
(1 P1
i=0 (
))i
t;t+i Yt+i
(1
))i
40
(
t;t+i =Pt+i )
t;t+i Yt+i
(
M Ct+i =Pt+i At+i Qt sjt Gjt+i
1 t;t+i =Pt+i )
:
(41)
Rewriting this equation yields Pjt? Pt
Qt
sjt Gjt
Qt 1 11+
=
Et
P1
i=0 (
Et
P1
i=0 (
t;t+i Pt
Yt+i t;t+i Yt
))i
(1
))i
(1
Qt+i =Qt M Ct+i Pt+i At+i Qt+i Gjt+i =Gjt
Pt+i
t;t+i Pt
Yt+i t;t+i Yt
1
:
(42)
Pt+i
The multi-period growth rate of the cohort e¤ect relative to the experience e¤ect corresponds to Qt+i =Qt qt+i = Gjt+i =Gjt gt+i
qt+1 ; gt+1
for i > 0, and equals unity for i = 0. Hence, this growth rate is independent of the index j, because when going forward in time, …rms are subject to the same experience e¤ect. Thus, we can rewrite the equation (42) according to Pjt? Pt
Qt
sjt Gjt
1 11+
=
Qt
Nt ; Dt
where the numerator Nt is given by Nt = Et
1 X
( (1
i=0
))i
t;t+i
t;t+i Pt
Yt+i Yt
M Ct+i Pt+i At+i Qt+i
Pt+i
qt+i gt+i
qt+1 gt+1
:
The numerator evolves recursively as shown by equation (11). The denominator Dt also evolves recursively, and jointly this yields the recursive pricing equations (10)-(12).
A.3
First-Order Conditions to the Household Problem
The …rst-order conditions that belong to the household problem comprise the household’s budget constraint, a no-Ponzi scheme condition, the transversality condition, and the following equations: Wt = Pt
ULt UCt
=
t+1
t;t+1
t
UCt+1 UCt 1 + it
1 = Et
t;t+1 t+1
1 = Et [
t;t+1 (rt+1
+1
d)] :
Here, we denote by U (:) the period utility function. Our assumption that U (Ct ; Lt ) = ([Ct V (Lt )]1
1)=(1
) implies UCt = Ct V (Lt )1 ULt = Ct1
V (Lt )
VLt ;
where UCt = @U (Ct ; Lt )=@Ct and VLt = @V (Lt )=@Lt . 41
A.4
Recursive Evolution of the Price Level
Plugging the weighted average price of a cohort, equation (13), into the price level, equation (14), yields that Pt1
? 1 t;t Pt;t )
= (
+
1 X
)s
(1
s=1
"
(1
)
s 1 X
k
1 ? t k;t Pt s;t k )
(
s
+
1 ? t s;t Pt s;t s )
(
k=0
Telescoping the sums yields: Pt1
? 1 t;t Pt;t )
= ( + (1
)1 (1
)(
1 ? t;t Pt 1;t )
+ (
+ (1
)2 (1
)(
? 1 t;t Pt 2;t )
+ (1
1 ? t 1;t Pt 1;t 1 )
) (
? 1 t 1;t Pt 2;t 1 )
2
+
(
? 1 t 2;t Pt 2;t 2 )
+ ::: : Collecting optimal prices that were set at the same date in square brackets yields: Pt1
= 1 t;t
? 1 (Pt;t )
+ [ (1
)]
+ (1 (Pt?
1 t 1;t
) (Pt?
)(1 1 1;t 1 )
1 1;t )
+ (1
)(Pt?
+ (1
) (Pt?
)(1
1 2;t )
+ (1
)2 (Pt?
1 2;t 1 )
+ (1
)(Pt?
1 3;t )
1 3;t 1 )
+ ::: + :::
+ ::: : Using equation (15) and the de…nition of pet in equation (17), we can replace the terms in curly brackets in the previous equation by pet . This yields
Pt1
= (
? 1 t;t Pt;t )
1 + (1
+ [ (1
)]1 (
+ [ (1
)]2 (
(pet )
)
? 1 t 1;t Pt 1;t 1 )
t
? 2;t Pt
2;t
1 2)
1
1
1 + (1
)
1 + (1
)
(pet 1 )
1
(pet 2 )
1
1 1
+ ::: :
Rearranging the previous equation yields Pt1
=(
? 1 t;t Pt;t )
+ (1
)(pet )
? 1 t 1;t 1 Pt 1;t 1 )
+ (1
)(
1 t 1;t )
+ (1
)(
? 1 t 2;t 1 Pt 2;t 2 )
(
1
+ (1
)(pet 1 )
+ (1 )(pet 2 )
1
+ :::
1
:
The term in curly brackets in the previous equation corresponds to Pt1 1 , which yields the price level equation (16) in the main text. 42
#
:
A.5
Equilibrium De…nition
We are now in a position to de…ne the market equilibrium: De…nition 1 An equilibrium is a state-contingent path for f(Pjt ; Ljt ; Kjt ) for j 2 [0; 1], Wt ; rt ; it ; Ct ; Kt+1 ; Lt ; Bt ; Tt g1 t=0 such that
1. the …rms’choices fPjt ; Ljt ; Kjt g1 t=0 maximize pro…ts for all j 2 [0; 1], given the price adjustment frictions,
2. the household’s choices fCt ; Kt+1 ; Lt ; Bt g1 t=0 maximize expected household utility, 3. the government ‡ow budget constraint holds each period, and 4. the markets for capital, labor, …nal and intermediate goods and government bonds clear, given the initial values B 1 (1 + i 1 ); K0 ; Pj; 1 , and A 1 Q
A.6
1 sj;
1
Gj; 1 , with j 2 [0; 1].
Aggregate Technology and Aggregate Productivity
To derive the aggregate technology, we combine …rms’technology to produce the di¤erentiated product in equation (2) with product demand Yjt =Yt = (Pjt =Pt ) Qt =Qt Gjt
Yt At Qt
Pjt Pt
sjt
=
Kjt Ljt
1
to obtain
1
Ljt
Ft :
Integrating over all …rms with j 2 [0; 1], using labor market clearing, Lt =
R1 0
Ljt dj, and
the fact that optimizing …rms maintain the same (and hence the aggregate) capital labor ratio yields Yt At Qt
Z
1
Qt =Qt Gjt
0
Pjt Pt
sjt
1
dj = Kt
1
1
Lt
Ft :
Rearranging this equation and de…ning the (inverse) endogenous component of aggregate productivity as in equation (19) in the main text yields the aggregate technology (18). To derive the recursive representation of
t
shown in equation (20), we rewrite equa-
tion (19) according to t
Pt
=
Z
1
qt gt
0
qt gt
sjt +1
(Pjt )
dj;
sjt +1
using the processes describing the evolution of Qt and Gjt . As for the price level, we proceed with the aggregation in two steps. First, we aggregate the optimal prices of all …rms operating within a particular cohort. Second, we aggregate all cohorts in the economy. To this end, we rewrite
t =Pt t
Pt
=
in the previous equation according to
1 X
(1
s=0
43
)s b t (s);
(43)
using b t (s) =
8
0) and the e¢ cient allocation, combining this equation with
pet =
Under sticky prices ( equation (11) implies Nt = 1 + (1 pet
)Et
"
e Yt+1 e t;t+1 Yte
t+1 t;t+1
52
qt+1 gt+1
pet+1 pet
Nt+1 pet+1
#
:
(87)
Furthermore, equation (12) implies Dt = 1 + (1
"
)Et
#
1
e Yt+1 e t;t+1 Yte
t+1
(88)
Dt+1 :
t;t+1
Firms hit by a -shock in period t in the sticky-price economy choose the same optimal relative price as …rms receiving a -shock in period t in the ‡exible-price economy, i.e., ? =Pt = Nt =Dt = pet or equivalently Nt =pet = Dt , if it holds that Pt;t
pet+1 pet
qt+1 gt+1
t+1 t;t+1
(89)
= 1;
which follows from comparing the equations (87) and (88). To show that equation (89) holds under the optimal in‡ation rate stated in proposition 2, we lag this equation by one period and rearrange it to obtain t
pet = pet
t 1;t
1
gt : qt
Combining this equation with equation (17) implies that the optimal in‡ation rate as de…ned in equation (28) satis…es equation (89). Establishing (2): To show that, under the optimal in‡ation rate, …rms that are subject to a Calvo shock in period t and hence can adjust their price do not …nd it optimal to change their price, we need to establish that Pt?
? ? t k;t Pt k;t k ;
=
k;t
(90)
for all k > 0. Dividing this equation by the (optimal) aggregate price level Pt? the result from step (1), i.e.,
? =Pt? Pt;t
Pt? k;t = Pt? k
pet ,
=
and using
we obtain
Pt? k;t Pt? k
? t k;t
k
k
=
? e t k;t pt k :
Using equation (15), we can rewrite the previous equation as ? Pt;t Pt?
qt gt
qt gt
? Again using Pt;t =Pt? = pet and that
pet pet k
qt gt
qt gt
t k;t
Pt? = Pt? k
k+1 k+1
=
k+1
Qk
j=1
k+1
? e t k;t pt k :
t k+j 1;t k+j
further delivers !
? t
? t+1 k
? t 1;t
? t k;t+1 k
= 1:
Rewriting the previous equation as ? t ? t 1;t
qt pet gt pet 1
? t 1 ? t 2;t 1
qt gt
1 1
pet pet
1
? t+1 k
2
? t k;t+1 k
53
qt+1 gt+1
k k
pet+1 k pet k
!
=1
shows that each term in parenthesis is equal to unity under the optimal in‡ation rate, which follows from equation (89). This establishes that …rms that can adjust their price maintain the indexed price as given by equation (90). Establishing (3): We can establish the fact that the condition 2 causes initial prices to re‡ect initial relative productivities as follows. The pricing equations (10)-(12) imply under ‡exible prices and no markup distortion that Pjt? Pt
Qt
sjt Gjt
=
Qt
M Ct : Pt At Qt
For a …rm hit by a -shock in period t, this equation yields pet =
M Ct : Pt At Qt
Combining both previous equations yields Pjt? = Pt
Qt sjt Gjt
Qt
pet :
Plugging this equation into the aggregate price level, Pt1 1=
Z
0
1
sjt Gjt
Rewriting this equation and using pet = 1=
E
1
Qt Qt
=
e t
R1 0
Pjt1
dj, yields
(pet )1 dj:
yields equation (25) for t =
1.
Discontinuity of the Optimal In‡ation Rate
This appendix compares the optimal in‡ation rate in an economy with -shocks ( > 0) to the economy in the absence of such shocks ( = 0). We refer to the …rst economy as the -economy and to the latter as the 0-economy. Comparing these two economies is not as straightforward as it might initially appear: even if both economies are subject to the same fundamental shocks (at ; qt ; gt ; t ), the e¢ cient allocation displays a discontinuity when considering the limit ! 0. The discontinuity arises because aggregate productivity growth in the -economy is driven by at qt , while it is driven by at gt in the 0-economy.
To properly deal with this issue, we construct a -economy whose e¢ cient aggregate allocation (consumption, hours, capital) is identical to the e¢ cient aggregate allocation in the 0-economy.37 We then compare the optimal in‡ation rates in these two economies and show that the optimal in‡ation rate for the -economy di¤ers from the optimal in‡ation rate for the 0-economy, even for the limit 37
! 0.
The two economies do of course di¤er in their underlying …rm-level dynamics.
54
Let at , qt , gt denote the productivity disturbances in the -economy and let A 1 Gj; 1 Q for j 2 [0; 1] denote the initial distribution of …rm productivities. This, together with the process f
1 jt gt=0
Gjt , and Qt
sjt
for all j 2 [0; 1], determines the entire state-contingent values for At , Qt ,
for all j 2 [0; 1] and all t
0.
Next, consider the 0-economy and suppose it starts with the same initial capital stock as the -economy. For the 0-economy, we normalize Q0t
1 for all j 2 [0; 1] and all t
sjt
and then set the initial …rm productivity distribution in the 0-economy equal to that in the -economy by choosing the initial conditions A0 1 = A 1 ; G0j;
1
= Gj; 1 Q
1 sj;
1
:
Finally, let the process for common TFP in the 0-economy be given by Z
A0t = At
1
1
1
Qt
0
1
dj
sjt Gjt
Z
1
G0jt
1
1 1
dj
;
0
where G0jt is generated by an arbitrary process gt0 , e.g., gt0 = gt . In this setting, it is easily veri…ed that aggregate productivity associated with the e¢ cient allocation, de…ned as At Qt =
e t
= At Qt
Z
1
1
Gjt Qt
sjt =Qt
1
1
dj
;
0
is the same in the -economy and the 0-economy.38 We then have the following result: Proposition 6 Under the assumptions stated in this section, the e¢ cient allocations in the two economies, the -economy and the 0-economy, satisfy Ct = Ct0 ; Lt = L0t ; Kt = Kt0 for all t
0 and all possible realizations of the disturbances.
Proof. Since At Qt =
e; t
= A0t Q0t =
e;0 t
for all t, it follows from the planner’s problem
(23)-(24) and the fact that the initial capital stock is identical that both economies share the same e¢ cient allocation. The following proposition shows that (generically) the optimal in‡ation rate discontinuously jumps when moving from the 0-economy to the -economy, even if both economies are identical in terms of their e¢ cient aggregate dynamics:39 38
The fact that At Qt = et is equal to aggregate productivity in the e¢ cient allocation follows from equations (24) and (25). 39 Recall that the optimal in‡ation rates implement the e¢ cient aggregate allocations in these economies.
55
1 sj;
1
Lemma 3 Under the assumptions stated in this section and provided conditions 1 and 2 hold, the optimal in‡ation rate in the 0-economy is
?;0 t
= 1 for all t. The optimal in‡ation
rate in the -economy is given by equation (28); in particular, for gt = g and qt = q, and in the absence of price indexation, the optimal rate of in‡ation in the -economy satis…es limt!1
?; t
= g=q.
Proof. The results directly follow from proposition 2 and lemma 1. The previous result illustrates the fragility of the optimality of strict price stability in standard sticky-price models, once non-trivial …rm-level productivity trends are taken into account. Moreover, in combination with proposition 6, the result shows that two economies that can be identical in terms of their aggregate e¢ cient allocations may require di¤erent in‡ation rates for implementing these allocations.
F
Proof of Proposition 3
Under the assumptions stated in the proposition, it is straightforward to show that the relative price distortion ( ) and the markup distortion ( ), which are de…ned in equations (63), (64) and (65), are inversely proportional to each other, ( ) = 1= ( ): As a result, the solution of L determined in equation (72) in appendix A.9 simpli…es to 1 (1 + )
L=
1=
;
because L( ) = 1 and, therefore, L no longer depends on the steady-state in‡ation rate . This result implies that L(1) = L(
?
), as stated in proposition 3.
In this case, the solutions for capital and consumption, equations (73) and (74), imply k( ) = ( ) c( ) = ( )
1 1
1
e
( 1
1 + d)
L;
1
1
(
e
1 + d)1
V (L) VL
;
where we explicitly indicate that steady-state capital and consumption depend on
.
Comparing steady-state consumption for the policy implementing the optimal in‡ation rate
?
and the alternative policy implementing strict price stability in economies without
price indexation yields (1) ( ?)
c(1) = c( ? ) 56
:
Equations (63) and (64) imply that the relative price distortion (
?
) = 1. This yields
c(1) = (1) ; c( ? ) e
= =
(1) 1
(1
)(g=q) (1 )
1
1
1 1
1
(1 (1
) (g=q) )(g=q)
1 1
!
;
which is the expression in proposition 3. To show that c(1)=c(
?
)
1, note that c(1)=c(
show that the inequality holds strictly, c(1)=c( of c(1)=c( @ @(g=q)
?
) with respect to g=q. This yields
c(1) c( ? )
=
c(1) c( ? )
(1 ) 2 (g=q)
1
?
?
) = 1, if g = q and hence
if 1
< 1, and g=q > (1
= 1. To
) < 1, for g 6= q, we take the derivative
(1
1 (g=q) ) (g=q) 1 [1 (1
Terms in square brackets are positive, because we have assumed that (1 (see equation (7)),
?
)(g=q) )(g=q)
1
1]
:
0 and thus g=q < 1. The derivative is strictly negative if 1
(g=q) < 0
and thus g=q > 1. The derivative is zero if g=q = 1.
G
Data Appendix
G.1
LBD Database
We use data from 1986 to 2013 dropping observations of establishments that were present already in the sample in 1976 for which age information is not available. We only consider establishments with at least one paid employee and truncate employment observations above the 99% percentile in a given industry and year. This leaves us with 147 million establishment-employment observations in our estimation sample. To improve the consistency of the mapping from SIC codes to NAICS codes, we follow the same establishments over the SIC-NAICS changeover to reverse engineer proper SIC codes for the considered industry z. Using this procedure, only about 0.2 million observations cannot be allocated to a NAICS code, because they have a coarse industry code under SIC. Since their employment share is negligible (0.02%), this should not a¤ect our estimates.
57
G.2
Sectoral Disaggregation and Sector Weights
z
and
e e z=
We use the value added series of the BEA GDP-by-Industry data for 71 industries and focus on the 65 private industries.40 To compute the sectoral trend growth rate
e z
entering lemma 2, we use the chain-type
quantity indexes for value added by industry for the years 1976 to 2013, which is the time span for which the LBD data is available. For a few industries (retail trade, hospitals, nursing and residential care facilities), data is only available from 1997 onwards. For these industries, we use data for the period 1997 to 2013. To compute the aggregate trend growth rate
e
entering lemma 2, we use the chain-
type quantity index for private industries for the years 1976 to 2013. To compute expenditure shares
z
for z = 1; : : : Z, we use the expenditure shares as
implied by the GDP statistics for the year 2013. To compute time-varying expenditure shares
zt
for z = 1; : : : Z, we use the expen-
diture shares as implied by the GDP statistics for the respective year. For a few sectors (retail trade, hospitals, nursing and residential care facilities), expenditure shares are not available for the period 1976 to 1996. We impute these shares using the distribution that we observe in 1997. Table 2 below reports how we map the LBD NAICS codes into the 65 BEA private industries (z = 1; 2; : : : 65). Table 2: BEA-NAICS Mapping z
BEA Code BEA Title
Related 2007 NAICS Codes
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
111CA 113FF 211 212 213 22 23 321 327 331 332 333 334 335 3361MV 3364OT 337 339 311FT 313TT
111, 112 113, 114, 115 211 212 213 221 230, 233 321 327 331 332 333 334 335 3361, 3362, 3363 3364, 3365, 3366, 3369 337 339 311, 312 313, 314
Farms Forestry, …shing, and related activities Oil and gas extraction Mining, except oil and gas Support activities for mining Utilities Construction Wood products Nonmetallic mineral products Primary metals Fabricated metal products Machinery Computer and electronic products Electrical equipment, appliances, and components Motor vehicles, bodies and trailers, and parts Other transportation equipment Furniture and related products Miscellaneous manufacturing Food and beverage and tobacco products Textile mills and textile product mills Continued on next page
40
The data is available at http://www.bea.gov/industry/gdpbyind_data.htm and was retrieved on August 24, 2016.
58
Table 2— continued from previous page z
BEA Code BEA Title
Related 2007 NAICS Codes
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
315AL 322 323 324 325 326 42 441 445 452 4A0 481 482 483 484 485 486 487OS 493 511 512 513 514 521CI 523 524 525 531 532RL 5411 5415 5412OP 55 561 562 61 621 622 623 624 711AS 713 721 722 81
315, 316 322 323 324 325 326 42 441 445 452 442, 443, 444, 446, 447, 448, 451, 453, 454 481 482 483 484 485 486 487, 488, 492 493 511 512 513, 515, 517 514, 518, 519 521, 522 523 524 525 531 532, 533 5411 5415 5412, 5413, 5414, 5416, 5417, 5418, 5419 55 561 562 611 621 622 623 624 711, 712 713 721 722 811, 812, 813, 814
G.3
Apparel and leather and allied products Paper products Printing and related support activities Petroleum and coal products Chemical products Plastics and rubber products Wholesale trade Motor vehicle and parts dealers Food and beverage stores General merchandise stores Other retail Air transportation Rail transportation Water transportation Truck transportation Transit and ground passenger transportation Pipeline transportation Other transportation and support activities Warehousing and storage Publishing industries, except internet (includes software) Motion picture and sound recording industries Broadcasting and telecommunications Data processing, internet publishing, and other information services Federal Reserve banks, credit intermediation, and related activities Securities, commodity contracts, and investments Insurance carriers and related activities Funds, trusts, and other …nancial vehicles Real estate Rental and leasing services and lessors of intangible assets Legal services Computer systems design and related services Miscellaneous professional, scienti…c, and technical services Management of companies and enterprises Administrative and support services Waste management and remediation services Educational services Ambulatory health care services Hospitals Nursing and residential care facilities Social assistance Performing arts, spectator sports, museums, and related activities Amusements, gambling, and recreation industries Accommodation Food services and drinking places Other services, except government
Sectoral Results for the Year 2007
The following table reports for the year 2007 a set of descriptive statistics and the regression outcomes.
59
Table 3: Descriptive Statistics and Regression Results, LBD, 2007
z
J
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
114200 25900 6900 6600 10200 21100 687700 15700 16700 4800 57100 24700 13400 5800 7900 4300 20000 28200 27700 9100 9300 4900 31500 2200 13100 13800 400800 118400 130900 44900 741800 5400 1200
P
j Lj
642400 203400 89700 172500 213200 561900 5490800 448600 402800 366200 1297500 874100 719500 334200 725700 352600 373600 467900 1235800 248800 175100 372900 499000 80100 626800 735600 4596200 1695500 2531700 2565300 6797900 217700 49000
Descriptive statistics
Linear speci…cation Linear-quadratic speci…cation E (ln Lj ) SD (ln Lj ) E (sj ) SD (sj ) 100 b 100 SE (b) R2 (in %) 100 b e 100 SE b e 100 b e 100 SE b e 1.06 1.27 1.48 2.31 1.89 2.12 1.34 2.42 2.34 3.19 2.24 2.51 2.67 2.82 3.05 2.75 1.89 1.76 2.50 2.04 1.96 3.49 1.84 2.22 2.72 3.05 1.63 1.93 1.88 2.76 1.69 2.08 2.27
1.03 1.13 1.33 1.36 1.47 1.48 1.14 1.42 1.33 1.67 1.35 1.47 1.64 1.65 1.82 1.83 1.36 1.34 1.58 1.51 1.35 1.48 1.28 1.52 1.56 1.49 1.22 1.20 1.41 1.62 1.03 1.67 1.68
11.33 11.36 12.88 14.84 9.23 18.27 10.10 14.69 16.26 20.08 17.21 18.59 15.47 17.82 16.02 14.54 13.41 14.38 15.59 15.08 10.47 22.14 16.25 17.54 17.60 17.63 12.60 12.90 11.20 11.13 11.51 10.93 9.92
6.89 9.13 10.73 11.88 9.24 12.10 9.22 12.01 13.53 14.82 12.92 13.11 11.94 13.18 12.72 12.75 10.93 11.53 13.09 12.17 10.96 14.62 12.01 14.29 13.74 12.48 10.38 10.37 10.09 9.62 9.86 9.97 8.39
1.80 2.27 0.48 2.22 1.73 2.67 3.27 3.52 2.72 4.48 3.27 3.29 4.11 4.68 3.52 5.40 4.00 3.37 4.85 4.25 2.77 4.20 2.94 3.70 3.60 3.27 3.29 2.56 4.15 5.66 1.58 5.46 4.83
0.04 0.08 0.15 0.14 0.16 0.08 0.01 0.09 0.07 0.15 0.04 0.07 0.11 0.15 0.16 0.20 0.08 0.07 0.07 0.12 0.13 0.13 0.06 0.21 0.09 0.10 0.02 0.03 0.04 0.07 0.01 0.22 0.56
1.45 3.35 0.15 3.75 1.18 4.79 7.04 8.92 7.63 15.82 9.84 8.63 8.91 14.03 6.03 14.12 10.40 8.42 16.14 11.66 5.02 17.11 7.58 12.18 10.04 7.49 7.80 4.88 8.78 11.34 2.31 10.63 5.86
Continued on next page
60
1.27 2.57 0:69 3.21 0.13 0:39 4.36 4.22 2.09 6.77 4.24 3.30 4.51 4.84 1.71 5.39 3.63 3.03 3.73 2.40 1.20 4.30 2.41 0.38 4.77 4.01 3.19 4.05 7.31 11.26 3.81 7.57 13.49
0.12 0.23 0.57 0.55 0.53 0.40 0.05 0.30 0.26 0.58 0.15 0.24 0.37 0.54 0.54 0.68 0.26 0.21 0.23 0.41 0.38 0.53 0.19 0.77 0.34 0.34 0.06 0.13 0.14 0.27 0.04 0.77 1.84
0.02 0:01 0.04 0:03 0.06 0.09 0:04 0:02 0.02 0:05 0:02 0.00 0:01 0.00 0.04 0.00 0.01 0.01 0.03 0.05 0.04 0.00 0.01 0.08 0:03 0:02 0.00 0:05 0:10 0:18 0:07 0:07 0:28
0.01 0.01 0.02 0.02 0.02 0.01 0.00 0.01 0.01 0.01 0.00 0.01 0.01 0.01 0.01 0.02 0.01 0.01 0.01 0.01 0.01 0.01 0.00 0.02 0.01 0.01 0.00 0.00 0.00 0.01 0.00 0.02 0.06
R2 (in %) 1.47 3.36 0.22 3.80 1.27 5.06 7.11 8.95 7.66 16.11 9.91 8.63 8.91 14.04 6.18 14.12 10.41 8.43 16.22 11.87 5.22 17.11 7.60 12.96 10.12 7.52 7.80 5.00 9.17 12.26 2.71 10.76 7.78
Table 3— continued from previous page z
J
34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
1600 103300 16300 2600 52000 14300 29800 19000 58000 23800 219400 79000 165700 7500 266100 62800 172800 461600 95100 50000 309800 20400 90100 505500 6900 72800 142500 39500 61500 55700 505500 691400
P
j
Descriptive statistics Lj
49800 1060600 398700 30900 810600 574500 711400 210400 1074800 427000 2164200 525300 1268900 58500 1157200 508400 889600 3495500 811900 2132900 5779000 309200 6748600 4373400 5018900 2780600 2075000 376700 1125900 1318400 8893400 4396100
Linear speci…cation Linear-quadratic speci…cation E (ln Lj ) SD (ln Lj ) E (sj ) SD (sj ) 100 b 100 SE (b) R2 (in %) 100 b e 100 SE b e 100 b e 100 SE b e 2.07 1.37 2.02 1.78 1.70 2.46 2.00 1.43 1.78 1.70 1.70 1.05 1.20 0.94 0.93 1.61 1.05 1.24 1.14 2.31 1.68 1.90 2.44 1.56 5.75 2.71 1.94 1.16 2.01 2.18 2.29 1.30
1.60 1.28 1.55 1.18 1.35 1.54 1.46 1.34 1.43 1.45 1.05 1.09 1.10 1.18 0.93 0.99 0.98 1.14 1.24 1.69 1.43 1.27 1.83 1.06 1.59 1.44 1.23 1.25 1.35 1.38 1.16 1.00
12.08 9.30 10.41 10.56 9.29 10.36 12.13 9.61 8.39 9.96 10.10 7.62 11.45 9.51 9.72 9.57 12.41 9.56 6.12 10.77 8.97 10.10 14.11 11.98 19.82 11.08 11.56 11.23 11.87 10.63 9.13 14.16
10.42 8.96 9.47 10.66 8.64 9.68 11.47 8.51 9.03 10.38 10.32 7.73 10.04 10.65 9.40 8.37 9.75 8.63 5.77 10.03 8.29 9.01 11.91 9.90 11.53 9.47 9.89 9.77 10.96 9.35 8.66 11.06
4.58 4.68 4.26 2.01 3.35 1.68 3.66 2.42 3.60 3.74 2.81 2.47 2.24 3.55 1.61 2.37 1.53 3.00 3.79 3.84 2.48 3.04 8.88 0.93 6.22 5.11 2.83 3.66 2.39 0.23 2.40 2.26
0.37 0.04 0.12 0.21 0.07 0.13 0.07 0.11 0.06 0.09 0.02 0.05 0.03 0.12 0.02 0.05 0.02 0.02 0.07 0.07 0.03 0.10 0.04 0.02 0.15 0.05 0.03 0.06 0.05 0.06 0.02 0.01
8.89 10.70 6.78 3.32 4.60 1.12 8.21 2.35 5.16 7.20 7.64 3.06 4.19 10.32 2.65 4.00 2.33 5.13 3.12 5.18 2.06 4.68 33.24 0.75 20.38 11.34 5.13 8.21 3.79 0.02 3.22 6.24
6.87 6.19 4.86 3.95 4.83 4.60 2.27 5.52 8.97 7.93 5.18 3.38 1.79 3.68 2.62 5.98 2.27 4.00 6.24 5.30 4.00 4.49 1:12 3.64 20.14 3.17 4.37 1.18 4.99 3.10 5.36 1.97
1.38 0.14 0.42 0.81 0.21 0.44 0.22 0.35 0.22 0.34 0.08 0.14 0.10 0.47 0.06 0.14 0.09 0.06 0.18 0.24 0.10 0.32 0.19 0.06 0.63 0.18 0.12 0.22 0.20 0.22 0.06 0.04
Notes: z denotes the industry number, see table 2; J denotes the number of establishments in industry z ;
61
P
j
0:07 0:05 0:02 0:06 0:05 0:09 0.04 0:11 0:18 0:13 0:08 0:03 0.01 0.00 0:03 0:13 0:02 0:04 0:11 0:05 0:05 0:05 0.29 0:09 0:39 0.06 0:05 0.08 0:08 0:09 0:11 0.01
0.04 0.00 0.01 0.03 0.01 0.01 0.01 0.01 0.01 0.01 0.00 0.01 0.00 0.01 0.00 0.00 0.00 0.00 0.01 0.01 0.00 0.01 0.01 0.00 0.02 0.01 0.00 0.01 0.01 0.01 0.00 0.00
R2 (in %) 9.06 10.82 6.80 3.55 4.70 1.46 8.35 2.80 6.22 7.81 8.07 3.12 4.20 10.33 2.75 5.08 2.37 5.19 3.34 5.26 2.15 4.78 35.34 1.24 25.95 11.48 5.25 8.53 4.08 0.36 3.72 6.24
Lj denotes total employment; E (ln Lj ) denotes
the mean of log employment; SD (ln Lj ) denotes the standard deviation of log employment; E (sj ) denotes the mean of establishment age; SD (sj ) denotes the standard deviation of establishment age; b denotes the estimated coe¢ cient obtained from the regression (35); SE (b) denotes the standard error of this coe¢ cient; R2 denotes the regression’s R-squared statistic; b e denotes the estimated coe¢ cient on age when augmenting the regression (35) with a regressor age squared; and b e denotes the estimated coe¢ cient on age squared.
62
